# LESSON 7.2 FACTORING POLYNOMIALS II

Size: px
Start display at page:

Transcription

1 LESSON 7.2 FACTORING POLYNOMIALS II LESSON 7.2 FACTORING POLYNOMIALS II 305

2 OVERVIEW Here s what you ll learn in this lesson: Trinomials I a. Factoring trinomials of the form x 2 + bx + c; x 2 + bxy + cy 2 You have already learned how to factor certain polynomials by finding the greatest common factor (GCF) and by grouping. In this lesson, you will learn techniques for factoring trinomials. Then you will see how to use factoring to solve certain equations. Trinomials II a. Factoring trinomials of the form ax 2 + bx + c, a 1, by trial-anderror b. Factoring trinomials of the form ax 2 + bx + c, a 1, by grouping c. Solving quadratic equations by factoring 306 TOPIC 7 FACTORING

3 EXPLAIN TRINOMIALS I Summary Factoring Polynomials of the Form x 2 + bx + c One way to factor a polynomial of the form x 2 + bx + c as a product of binomials is to use the FOIL method, but work backwards. Here s an example. The product of the first terms is x 2 x 2 3x 4 = (x + 4)(x + 4) The product of the last terms is 4 x 2 3x 4 = (x + 4)(x + 4) Try all the possible factorizations for which the product of the first terms is x 2 and the product of the last terms is 4. Since the product of the last terms is negative, one of the last terms is positive and the other is negative. Use the FOIL method to find factors whose inner and outer products add together to make 3x. 1. Make a chart of the possibilities for the binomial factors. These are shown in the table. possible factorizations (x + 4)(x 1) (x 4)(x + 1) (x + 2)(x 2) 2. Use the FOIL method to multiply the possible factorizations you listed in step (1). These are shown in the table. possible factorizations (x + 4)(x 1) = x 2 + 3x 4 (x 4)(x + 1) = x 2 3x 4 (x + 2)(x 2) = x Find the factorization that gives the original polynomial, x 2 3x 4. In the second row you see that x 2 3x 4 = (x 4)(x + 1). So the factorization is: x 2 3x 4 = (x 4)(x + 1). LESSON 7.2 FACTORING POLYNOMIALS II EXPLAIN 307

4 Answers to Sample Problems Sample Problems 1. Factor: x 2 + 3x + 2 a. List all the possible factorizations where: the product of the first terms is x 2 the product of the last terms is +2 Since the product of the last terms is positive, both of the last terms are positive or both are negative. possible factorizations (x + 1)(x + 2) = (x 1)(x 2) = b. Multiply the possible factorizations. Identify the factorization that gives the middle term +3x. possible factorizations b.x 2 + 3x + 2 (x + 1)(x + 2) = (x 1)(x 2) = x 2 3x +2 c. (x + 1)(x + 2) (in either order) c. Write the correct factorization. x 2 + 3x + 2 = 2. Factor: x 2 7x + 12 a. List all the possible factorizations where: the product of the first terms is x 2 the product of the last terms is +12 Since the product of the last terms is positive, both of the last terms are positive or both are negative. possible factorizations (x + 1)(x + 12) = (x 1)(x 12) = (x + 2)(x + 6) = a. 6 3 (x 2)(x ) = (x + )(x + 4) = (x 3)(x 4) = 308 TOPIC 7 FACTORING

5 b. Multiply the possible factorizations. Identify the factorization that gives the middle term 7x. Answers to Sample Problems possible factorizations (x + 1)(x + 12) = x x + 12 (x 1)(x 12) = (x + 2)(x + 6) = (x 2)(x ) = x 2 8x + 12 (x + )(x + 4) = x 2 + 7x + 12 (x 3)(x 4) = b. x 2 13x + 12 x 2 + 8x x 2 7x + 12 c. Write the correct factorization. x 2 7x + 12 = c. (x 3)(x 4) (in either order) 3. Factor: x 2 + x 2 a. List all the possible factorizations where: the product of the first terms is x 2 the product of the last terms is 2 Since the product of the last terms is negative, one of the last terms is positive and the other is negative. possible factorizations (x + 1)(x 2) = (x 1)( ) = a. x + 2 b. Multiply the possible factorizations. Identify the factorization that gives the middle term +1x. possible factorizations (x + 1)(x 2) = (x 1)(x + 2) = c. Write the correct factorization. x 2 + x 2 =. b. x 2 x 2 x + 2, x 2 + x 2 c. (x 1)(x + 2) (in either order) LESSON 7.2 FACTORING POLYNOMIALS II EXPLAIN 309

6 Answers to Sample Problems 4. Factor: x 2 + 2x - 2 a. List all the possible factorizations where: the product of the first terms is x 2 the product of the last terms is 2 Since the product of the last terms is negative, one of the last terms is positive and the other is negative. possible factorizations (x + 1)(x 2) = a. x + 2 (x 1)( ) = b. Multiply the possible factorizations. Identify the factorization that gives the middle term +2x. possible factorizations b. x 2 x 2 x + 2, x 2 + x 2 (x + 1)(x 2) = (x 1)( ) = c. Write the correct factorization. Neither of the possible factorizations gives the original polynomial, x 2 + 2x 2. So, x 2 + 2x 2 cannot be factored using integers. 310 TOPIC 7 FACTORING

7 TRINOMIALS II Summary Factoring Polynomials of the Form ax 2 + bx + c by Trial and Error You have learned how to factor trinomials of the form x 2 + bx + c, where b and c are integers. Notice that the coefficient of x 2 is 1. Now you will see how to factor trinomials of the form ax 2 + bx + c, where a, b, and c are integers. Notice that the coefficient of x 2 can be an integer other than 1. One way to factor a trinomial of the form ax 2 + bx + c as a product of binomials is by trial and error. Here s an example. Factor the trinomial 3x 2 14x 5 using trial and error. Notice that any factorization of this trinomial must look like this: 3x 2 14x 5 = (?x?)(?x?) The product of the x -terms must be 3x 2 and the product of the constants must be 5. Since the product of the constants is negative, one of the constants is positive and the other is negative. 1. Make a chart of the possibilities for the x-terms in the binomial factors and possibilities for the constant terms in the binomial factors. These are shown in the table below. x -terms constants 3x, x 1, 5 3x, x 5, 1 3x, x 1, 5 3x, x 5, 1 2. Use the values from step (1) to list possible factorizations. These are shown in the table below. x -terms constants possible factorizations 3x, x 1, 5 (3x + 1)(x 5) 3x, x 5, 1 (3x + 5)(x 1) 3x, x 1, 5 (3x 1)(x + 5) 3x, x 5, 1 (3x 5)(x + 1) LESSON 7.2 FACTORING POLYNOMIALS II EXPLAIN 311

8 3. Use the FOIL method to do the multiplication of the possible factorizations you listed in step (2). These are shown in the table below. x -terms constants possible factorizations 3x, x 1, 5 (3x + 1)(x 5) = 3x 2 14x 5 3x, x 5, 1 (3x + 5)(x 1) = 3x 2 + 2x 5 3x, x 1, 5 (3x 1)(x + 5) = 3x x 5 3x, x 5, 1 (3x 5)(x + 1) = 3x 2 2x 5 4. Find the factorization that equals the original polynomial, 3x 2 14x 5. You can see that the shaded row is 3x 2 14x 5. So the factorization is: 3x 2 14x 5 = (3x + 1)(x 5) Here s another example. Factor the trinomial 15x 2 16x + 4 using trial and error. Notice that any factorization of this trinomial must look like this: 15x 2 16x + 4 = (?x?)(?x?) The product of the x -terms must be 15x 2 and the product of the constant terms must be +4. Since the product of the last terms is positive, both of the last terms are positive or both are negative. 1. Make a chart of the possibilities for the x-terms in the binomial factors and possibilities for the constant terms in the binomial factors. These are shown in the table below. x -terms constants x, 15x 1, 4 x, 15x 2, 2 x, 15x 4, 1 x, 15x 1, 4 x, 15x 2, 2 x, 15x 4, 1 3x, 5x 1, 4 3x, 5x 2, 2 3x, 5x 4, 1 3x, 5x 1, 4 3x, 5x 2, 2 3x, 5x 4, 1 2. Use the values from step (1) to list possible factorizations. These are shown in the table that follows. 312 TOPIC 7 FACTORING

9 x -terms constants possible factorizations x, 15x 1, 4 (x + 1)(15x + 4) x, 15x 2, 2 (x + 2)(15x + 2) x, 15x 4, 1 (x + 4)(15x + 1) x, 15x 1, 4 (x 1)(15x 4) x, 15x 2, 2 (x 2)(15x 2) x, 15x 4, 1 (x 4)(15x 1) 3x, 5x 1, 4 (3x + 1)(5x + 4) 3x, 5x 2, 2 (3x + 2)(5x + 2) 3x, 5x 4, 1 (3x + 4)(5x + 1) 3x, 5x 1, 4 (3x 1)(5x 4) 3x, 5x 2, 2 (3x 2)(5x 2) 3x, 5x 4, 1 (3x 4)(5x 1) 3. Use the FOIL method to do the multiplication of the possible factorizations you listed in step (2). These are shown in the table. x -terms constants possible factorizations x, 15x 1, 4 (x + 1)(15x + 4) = 15x x + 4 x, 15x 2, 2 (x + 2)(15x + 2) = 15x x + 4 x, 15x 4, 1 (x + 4)(15x + 1) = 15x x + 4 x, 15x 1, 4 (x 1)(15x 4) = 15x 2 19x + 4 x, 15x 2, 2 (x 2)(15x 2) = 15x 2 32x + 4 x, 15x 4, 1 (x 4)(15x 1) = 15x 2 61x + 4 3x, 5x 1, 4 (3x + 1)(5x + 4) = 15x x + 4 3x, 5x 2, 2 (3x + 2)(5x + 2) = 15x x + 4 3x, 5x 4, 1 (3x + 4)(5x + 1) = 15x x + 4 3x, 5x 1, 4 (3x 1)(5x 4) = 15x 2 17x + 4 3x, 5x 2, 2 (3x 2)(5x 2) = 15x 2 16x + 4 3x, 5x 4, 1 (3x 4)(5x 1) = 15x 2 23x Find the factorization that equals the original polynomial, 15x 2 16x + 4. You can see that the shaded row is 15x 2 16x + 4. So the factorization is: 15x 2 16x + 4 = (3x 2)(5x 2) Here s another example. Factor the trinomial 3x 2 8x 5 using trial and error. Notice that any factorization of this trinomial must look like this: 3x 2 8x 5 = (?x?)(?x?) The product of the x -terms must be 3x 2 and the product of the constants must be 5. Since the product of the constants is negative, one of the constants is positive and the other is negative. LESSON 7.2 FACTORING POLYNOMIALS II EXPLAIN 313

10 1. Make a chart of the possibilities for the x -terms in the binomial factors and possibilities for the constant terms in the binomial factors. These are shown in the table below. x -terms constants 3x, x 1, 5 3x, x 5, 1 3x, x 1, 5 3x, x 5, 1 2. Use the values from step (1) to list possible factorizations. These are shown in the table below. x -terms constants possible factorizations 3x, x 1, 5 (3x + 1)(x 5) 3x, x 5, 1 (3x + 5)(x 1) 3x, x 1, 5 (3x 1)(x + 5) 3x, x 5, 1 (3x 5)(x + 1) 3. Use the FOIL method to do the multiplication of the possible factorizations you listed in step (2). These are shown in the table below. x -terms constants possible factorizations 3x, x 1, 5 (3x + 1)(x 5) = 3x 2 14x 5 3x, x 5, 1 (3x + 5)(x 1) = 3x 2 + 2x 5 3x, x 1, 5 (3x 1)(x + 5) = 3x x 5 3x, x 5, 1 (3x 5)(x + 1) = 3x 2 2x 5 4. Find the factorization that equals the original polynomial, 3x 2 8x 5. You can see that no row is 3x 2 8x 5. So, 3x 2 8x 5 cannot be factored using integers. Factoring Polynomials of the Form ax 2 + bx + c by Grouping Another way to factor a trinomial of the form ax 2 + bx + c is by grouping. Remember how to multiply binomials using the FOIL method. (x + 2)(3x + 1) = 3x 2 + x + 6x + 2 = 3x 2 + 7x + 2 To factor 3x 2 + 7x + 2, we go the other way. We first write 3x 2 + 7x + 2 using two x-terms, like this: 3x 2 + x + 6x + 2 Now, factor 3x 2 + x + 6x + 2 by grouping: 314 TOPIC 7 FACTORING

11 1. Factor each term. 3x 2 = 3 x x x = x 6x = 2 3 x 2 = 2 2. Group terms with common factors. = (3x 2 + x) + (6x + 2) 3. Factor out the GCF in each grouping. = x(3x + 1) + 2(3x + 1) 4. Factor out the binomial GCF = (3x + 1)(x + 2) of the polynomial. 5. Check your answer. Is (3x + 1)(x + 2) = 3x 2 + 7x + 2? Is 3x 2 + 7x + 2 = 3x 2 + 7x + 2? Yes. In order to use grouping to factor this trinomial, you had to find two integers whose sum was 7 and whose product was 6. To factor a trinomial of the form ax 2 + bx + c, you need to find two integers whose sum is b and whose product is ac. Then you can split the x-term into two terms and factor by grouping. For example, to factor 6x 2 + 7x + 2 by grouping: 1. Make a chart of possible pairs of integers product is 6 2 = 12. possibilities product sum 1, , , Identify the numbers that work. Here, the last choice works since = 7 and 3 4 = 12. Notice that the chart doesn t include negative factors of 12. Can you see why not? Since the product of the two numbers has to be +12, if one factor is negative, both would have to be negative. But since the sum of the integers needs to be +7, a positive number, you know both factors can t be negative. 3. Rewrite the trinomial. 6x 2 + 7x + 2 = 6x 2 + 3x + 4x Group the terms. = (6x 2 + 3x ) + (4x + 2) 5. Factor out the GCF in each grouping. = 3x (2x + 1) + 2(2x + 1) 6. Factor out the binomial = (2x + 1)(3x + 2) GCF of the polynomial. 7. Check your answer. Is (2x + 1)(3x + 2) = 6x 2 + 7x + 2? Is 6x 2 + 4x + 3x + 2 = 6x 2 + 7x + 2? Yes. LESSON 7.2 FACTORING POLYNOMIALS II EXPLAIN 315

13 Sample Problems Answers to Sample Problems 1. Use trial and error to factor the polynomial 35x x + 6. a. Write possible x-terms whose product is 35x 2 and write possible constant terms whose product is 6. b. List the possible factorizations. c. Multiply the possible factorizations. x-terms constants possible factorizations x, 35x 1, 6 (x + 1)(35x + 6) = 35x x + 6 x, 35x 2, ( )( ) = x, 35x 3, 2 (x + 3)(35x + 2) = 35x x + 6 x, 35x 6, ( )( ) = x, 35x, 6 (x 1)(35x 6) = 35x 2 41x + 6 x, 35x 2, ( )( ) = x, 35x 3, 2 (x 3)(35x 2) = 35x 2 107x + 6 x, 35x 6, ( )( ) = 5x, 7x 1, 6 (5x + 1)(7x + 6) = 5x, 7x 2, ( )( ) = 35x x + 6 5x, 7x 3, 2 (5x + 3)(7x + 2) = 35x x + 6 5x, 7x 6, (5x + 6)(7x + 1) = 5x, 7x 1, ( )( ) = 5x, 7x, 3 ( )( ) = 5x, 7x 3, 2 (5x 3)(7x 2) = 35x 2 31x + 6 5x, 7x, 1 ( )( ) = d. Write the correct factorization. 35x x + 6 = ( in either order ) a., b., c. 3, (x + 2)(35x + 3) = 35x x + 6 1, (x + 6)(35x + 1) = 35x x , (x 2)(35x 3) = 35x 2 73x + 6 1,(x 6)(35x 1) = 35x 2 211x x x + 6 3, (5x + 2)(7x + 3) 1, 35x x + 6 6, (5x 1)(7x 6) = 35x 2 37x + 6 2, (5x 2)(7x 3) = 35x 2 29x + 6 6, (5x 6)(7x 1) = 35x 2 47x + 6 (x + 2)(35x + 3) 2. Use trial and error to factor 4x 2 4x 15. a. Write possible x-terms whose product is 4x 2 and the possible constant terms whose product is 15. b. List the possible factorizations. c. Multiply the possible factorizations. LESSON 7.2 FACTORING POLYNOMIALS II EXPLAIN 317

14 Answers to Sample Problems x-terms constants possible factorizations x, 4x 1, 15 (x + 1)(4x 15) = 4x 2 11x 15 2x, 2x 1, 15 (2x + 1)(2x 15) = 4x 2 28x 15 x, 4x 3, 5 (x + 3)(4x 5) = 4x 2 + 7x 15 2x, 2x 3, 5 (2x + 3)(2x 5) = 4x 2 4x 15 x, 4x 5, 3 (x + 5)(4x 3) = 4x x 15 a., b., c. 1, x +15, 4x 1, 4x x 15 1, 2x +15, 2x 1, 4x x 15 15,x 1, 4x + 15, 4x x 15 15, 2x 1, 2x + 15, 4x x 15 3, x 3, 4x + 5, 4x 2 7x 15 3, 2x 3, 2x + 5, 4x 2 + 4x 15 5, x 5, 4x + 3, 4x 2 17x 15 5, 2x 5, 2x + 3, 4x 2 4x 15 1, x 15, 4x + 1, 4x 2 59x 15 1, 2x 15, 2x + 1, 4x 2 28x 15 d. 2x 5, 2x + 3 (in either order) 2x, 2x 5, 3 (2x + 5)(2x 3) = 4x 2 + 4x 15 x, 4x 15, ( )( ) = 2x, 2x 15, ( )( ) = x, 4x 1, ( )( ) = 2x, 2x 1, ( )( ) = x, 4x, 5 ( )( ) = 2x, 2x, 5 ( )( ) = x, 4x, 3 ( )( ) = 2x, 2x, 3 ( )( ) = x, 4x 15, ( )( ) = 2x, 2x 15, ( )( ) = d. Write the correct factorization. 4x 2 4x 15 = ( )( ) 3. Use grouping to factor 6x x + 4. a. Make a chart of pairs of integers whose product is 6 4 = 24. possibilities product sum 1, , , , b. Identify the two integers whose product is 24 and whose sum is 11. The two integers are 3 and 8. c. Rewrite the trinomial by 6x x + 4 = 6x 2 + 3x + 8x + 4 splitting the x-term. d. 6x 2 + 3x, 8x + 4 e. 3x, 4 f. (3x + 4) d. Group the terms. = ( ) + ( ) e. Factor out the GCF in each grouping. = (2x + 1) + (2x + 1) f. Factor out the binomial = (2x + 1)( ) GCF of the polynomial. 318 TOPIC 7 FACTORING

15 g. Check your answer. 4. Use grouping to factor 3x 2 4x 15. a. Make a chart of pairs of integers whose product is 3 ( 15) = 45. possibilities product sum 1, , , 45 1, 45, 15 45, 45 b. Identify the two integers whose The two integers are and. product is 45 and whose sum is 4. c. Rewrite the trinomial 3x 2 4x 15 = 3x 2 + x + x 15 by splitting the x-term. d. Group the terms. = ( ) + ( ) e. Factor out the GCF in each grouping. = ( ) + ( ) f. Factor out the binomial GCF of the polynomial. = ( )( ) g. Check your answer. 5. Solve this quadratic equation for x by factoring: 8x 2 = 26x + 45 a. Write the equation in standard form. 8x 2 26x 45 = 0 b. Factor the left side. ( )( ) = 0 c. Use the Zero Product Property. 4x + 5 = or 2x 9 = d. Finish solving for x. 4x = 5 or 2x = 9 Answers to Sample Problems g. (2x + 1)(3x + 4) = 2x (3x ) + 2x (4) + 1(3x ) + 1(4) = 6x 2 + 8x + 3x + 4 = 6x x + 4 a. 12 9, 4 45, 44 3, 12 5, 9, 4 b. 5, 9 (in either order) c. 5, 9 (in either order) d. (3x 2 9x) + (5x 15) or (3x 2 + 5x) (9x +15) e. 3x(x 3) + 5(x 3) or x(3x + 5) 3(3x + 5) f. (x 3)(3x + 5) in either order g. (x 3)(3x + 5) = 3x 2 + 5x 9x 15 = 3x 2 4x 15 b. 4x + 5, 2x 9 (in either order) c. 0, d., e. Is 8 2 = ? e. Check your answer. x = or x = Is 8 = ? Is = ? Is = 2 2? Yes. 9 2 Is 8 2 = ? Is 8 = ? Is 162 = ? Is 162 = 162? Yes. LESSON 7.2 FACTORING POLYNOMIALS II EXPLAIN 319

16 Answers to Sample Problems EXPLORE Sample Problems On the computer you used overlapping circles to help find the GCF of a collection of monomials. You used a table to help factor polynomials. Below are some additional problems. 1. Use overlapping circles to find the GCF of 3x and 9xy 3. a. Factor each monomial. b. Write the factorizations in the overlapping circles. 3x = 3 x 9xy 3 = x y y y 1 3 x ( 1) 3 y y y c. 3x c. Find the GCF from the GCF = overlapping circles. a. 1 x x y x y Factor: x 2 y xy a. Factor each monomial x 2 y = xy = b. 1 xy b. Find the GCF of x 2 y GCF = c. 1 xy(x 3) 5 and xy. c. Factor the polynomial x 2 y 3 5 xy. ( )( ) 320 TOPIC 7 FACTORING

17 3. Find the GCF of the polynomials below. A: 22x 2 z + 22yz Answers to Sample Problems B: 11x xy C: 2x 2 + 2y a. Factor each polynomial. 22x 2 z + 22yz = 2 11 z (x 2 + y) 11x xy = 2x 2 + 2y = a. 11 x (x 2 +y) 2 (x 2 +y) b. Finish writing the factorizations in the overlapping circles. b. A z 11 2 x 2 + y 1 x B C 4. A trinomial with a missing constant term has been partially factored in the table below. Complete the table and write the polynomial and its factorization. a. What times x gives 7x? Use this to fill in box a. x b b. What times 3x gives 9x? 3x 3x 2 9x Use this to fill in box b. c c. Multiply boxes a and b. Use this to fill in box c. d. Write the polynomial a 7x and its factorization. ( ) = ( ) + ( ) a., b., c. x 3 3x 3x 2 9x 7 7x 21 3x 2 2x 21 = (3x + 7)(x 3) d. LESSON 7.2 FACTORING POLYNOMIALS II EXPLORE 321

18 HOMEWORK Homework Problems Circle the homework problems assigned to you by the computer, then complete them below. Explain Trinomials I 1. Factor: x 2 + 7x Factor: y 2 + 9y Factor: x x Factor: z z Factor: x 2 5x Factor: a 2 15a Factor: x 2 x 6 8. Factor: x x Factor: x 2 4x Factor: y 2 + 3y Factor: x x Factor: a 2 9a + 14 Trinomials II 13. Factor: 2x x Factor: 3x x Factor: 4y 2 8y Factor: 3z 2 17z Factor: 15a 2 30a Solve for x by factoring: 6x 2 = 63 13x 19. Solve for x by factoring: 25x 2 + 5x = Factor: 4x 2 12x Factor: 13x x Factor: x 2 a Factor: x 2 + 2xy + y Factor: x 4 2ax 2 + a 2 Explore 25. Circle the monomial(s) below that might appear in the factorization of 3x 3 y 2 + 2x 2 y 3xy 3x 2 y 2x 2 y xy 3x 26. If the GCF of the terms of a polynomial is 4x 2 y 3, which of the monomials below could be terms in the polynomial? 4xy 3 8x 3 y 4 4x 2 y 3 4x Factor this polynomial using overlapping circles: 28. A trinomial with a missing constant term has been partially factored in the table below. Complete the table and write the polynomial and its factorization. 29. Complete the diagram below to find the GCF of the polynomials A, B, and C. A = 2xy 2 z 30. Factor this polynomial using overlapping circles: 1 2 x 2 y 2 + x 5x 5x 2 10x 4x C = 13x 3 x 3 y 3 3x 2 y 2 B = 52x 2 y x 2 y 2 2y TOPIC 7 FACTORING

19 APPLY Practice Problems Here are some additional practice problems for you to try. Trinomials I 1. Factor: x 2 5x 4 2. Factor: x 2 + 6x 5 3. Factor: x 2 15x Factor: x 2 11x Factor: x 2 8x Factor: x 2 9x Factor: x 2 7x Factor: x 2 13x Factor: x 2 8x Factor: x 2 7x Factor: x 2 15x Factor: x 2 11x Factor: x 2 10x Factor: x 2 6x Factor: x 2 7x Factor: x 2 5x Factor: x 2 4x Factor: x 2 10x Factor: x 2 5x Factor: x 2 2x Factor: x 2 7x Factor: x 2 9x Factor: x 2 4x Factor: x x Factor: x 2 2x Factor: x 2 + 9x Factor: x 2 7x Factor: x 2 6x 91 Trinomials II 29. Factor: 2x 2 7x Factor: 2x 2 9x Factor: 3x 2 19x Factor: 2x 2 3x Factor: 2x 2 x Factor: 3x 2 16x Factor: 2x 2 5x Factor: 2x 2 9x Factor: 2x 2 13x Factor: 2x 2 15x Factor: 3x 2 11x Factor: 12x 2 7x Factor: 10x 2 9x Factor: 6x 2 5x Factor: 6x 2 11x Factor: 9x 2 18x Factor: 8x 2 2x Factor: 6x 2 13x Factor: 9x 2 3x Factor: 4x 2 4x Factor: 36x 2 13x Factor: 30x 2 11x Factor: 5x 2 14xy 3y Factor: 4x 2 7xy 2y Factor: 3x 2 5xy 2y Factor: 6x 2 xy 12y Factor: 9x 2 3xy 2y Factor: 4x 2 4xy 3y 2 LESSON 7.2 FACTORING POLYNOMIALS II APPLY 323

20 Practice Test EVALUATE Take this practice test to be sure that you are prepared for the final quiz in Evaluate. 1. Factor: x 2 10x Circle the statement(s) below that are true. 10. The overlapping circles contain the factors of two binomials, A and B. Their GCF is (3u + 4v ). What are A and B? x 2 + 2x 1 = (x 1)(x 1) 5u 3u + 4v 3v x 2 + 2x 1 = (x + 2)(x 1) x 2 + 2x 1 = (x 1)(x + 1) x 2 + 2x 1 = (x + 1)(x + 1) x 2 + 2x 1 cannot be factored using integers 3. Factor: t 2 16t The polynomial 14xy + 21y 6x 2 9x can be grouped as two binomials: (14xy 6x 2 ) + (21y 9x). Find the GCF of the two binomials by factoring the polynomial using the overlapping circles below. 14xy 6x 2 21y 9x 4. Factor: r rt + 25t 2 5. Factor: 5x 2 + 8x 4 6. Factor: 27v 2 57v Factor: 4x x Finish factoring the trinomial 6x 2 7xy 3y 2 using the table below. y 8. Solve for x by factoring: 7x 2 5x 12 = 0 9. The overlapping circles contain the factors of three monomials, A, B, and C. Circle the true statements below. A B 2 x 6z 2 4z 3 2x 6x 2 3y 2 C Two factors of C are z and 2. B = 72xz The GCF of A and B is x. The GCF of A, B, and C is 4z. 324 TOPIC 7 FACTORING

### Multiplication of Polynomials

Summary 391 Chapter 5 SUMMARY Section 5.1 A polynomial in x is defined by a finite sum of terms of the form ax n, where a is a real number and n is a whole number. a is the coefficient of the term. n is

### LESSON 7.1 FACTORING POLYNOMIALS I

LESSON 7.1 FACTORING POLYNOMIALS I LESSON 7.1 FACTORING POLYNOMIALS I 293 OVERVIEW Here s what you ll learn in this lesson: Greatest Common Factor a. Finding the greatest common factor (GCF) of a set of

### LESSON 9.1 ROOTS AND RADICALS

LESSON 9.1 ROOTS AND RADICALS LESSON 9.1 ROOTS AND RADICALS 67 OVERVIEW Here s what you ll learn in this lesson: Square Roots and Cube Roots a. Definition of square root and cube root b. Radicand, radical

### When you square a binomial, you can apply the FOIL method to find the product. You can also apply the following rules as a short cut.

Squaring a Binomial When you square a binomial, you can apply the FOIL method to find the product. You can also apply the following rules as a short cut. Solve. (x 3) 2 Step 1 Square the first term. Rules

Adding and Subtracting Polynomials Polynomial A monomial or sum of monomials. Binomials and Trinomial are also polynomials. Binomials are sum of two monomials Trinomials are sum of three monomials Degree

### A-2. Polynomials and Factoring. Section A-2 1

A- Polynomials and Factoring Section A- 1 What you ll learn about Adding, Subtracting, and Multiplying Polynomials Special Products Factoring Polynomials Using Special Products Factoring Trinomials Factoring

### Algebra 2. Factoring Polynomials

Algebra 2 Factoring Polynomials Algebra 2 Bell Ringer Martin-Gay, Developmental Mathematics 2 Algebra 2 Bell Ringer Answer: A Martin-Gay, Developmental Mathematics 3 Daily Learning Target (DLT) Tuesday

### Never leave a NEGATIVE EXPONENT or a ZERO EXPONENT in an answer in simplest form!!!!!

1 ICM Unit 0 Algebra Rules Lesson 1 Rules of Exponents RULE EXAMPLE EXPLANANTION a m a n = a m+n A) x x 6 = B) x 4 y 8 x 3 yz = When multiplying with like bases, keep the base and add the exponents. a

### Topic 7: Polynomials. Introduction to Polynomials. Table of Contents. Vocab. Degree of a Polynomial. Vocab. A. 11x 7 + 3x 3

Topic 7: Polynomials Table of Contents 1. Introduction to Polynomials. Adding & Subtracting Polynomials 3. Multiplying Polynomials 4. Special Products of Binomials 5. Factoring Polynomials 6. Factoring

### Algebra 1: Hutschenreuter Chapter 10 Notes Adding and Subtracting Polynomials

Algebra 1: Hutschenreuter Chapter 10 Notes Name 10.1 Adding and Subtracting Polynomials Polynomial- an expression where terms are being either added and/or subtracted together Ex: 6x 4 + 3x 3 + 5x 2 +

### MathB65 Ch 4 VII, VIII, IX.notebook. November 06, 2017

Chapter 4: Polynomials I. Exponents & Their Properties II. Negative Exponents III. Scientific Notation IV. Polynomials V. Addition & Subtraction of Polynomials VI. Multiplication of Polynomials VII. Greatest

### Algebra I Polynomials

Slide 1 / 217 Slide 2 / 217 Algebra I Polynomials 2014-04-24 www.njctl.org Slide 3 / 217 Table of Contents Definitions of Monomials, Polynomials and Degrees Adding and Subtracting Polynomials Multiplying

### Algebra I. Polynomials.

1 Algebra I Polynomials 2015 11 02 www.njctl.org 2 Table of Contents Definitions of Monomials, Polynomials and Degrees Adding and Subtracting Polynomials Multiplying a Polynomial by a Monomial Multiplying

Algebra I Book 2 Powered by... ALGEBRA I Units 4-7 by The Algebra I Development Team ALGEBRA I UNIT 4 POWERS AND POLYNOMIALS......... 1 4.0 Review................ 2 4.1 Properties of Exponents..........

### Review Notes - Solving Quadratic Equations

Review Notes - Solving Quadratic Equations What does solve mean? Methods for Solving Quadratic Equations: Solving by using Square Roots Solving by Factoring using the Zero Product Property Solving by Quadratic

### UNIT 2 FACTORING. M2 Ch 11 all

UNIT 2 FACTORING M2 Ch 11 all 2.1 Polynomials Objective I will be able to put polynomials in standard form and identify their degree and type. I will be able to add and subtract polynomials. Vocabulary

### 5.3. Polynomials and Polynomial Functions

5.3 Polynomials and Polynomial Functions Polynomial Vocabulary Term a number or a product of a number and variables raised to powers Coefficient numerical factor of a term Constant term which is only a

### KEY CONCEPTS. Factoring is the opposite of expanding.

KEY CONCEPTS Factoring is the opposite of expanding. To factor simple trinomials in the form x 2 + bx + c, find two numbers such that When you multiply them, their product (P) is equal to c When you add

### MULTIPLYING TRINOMIALS

Name: Date: 1 Math 2 Variable Manipulation Part 4 Polynomials B MULTIPLYING TRINOMIALS Multiplying trinomials is the same process as multiplying binomials except for there are more terms to multiply than

### Can there be more than one correct factorization of a polynomial? There can be depending on the sign: -2x 3 + 4x 2 6x can factor to either

MTH95 Day 9 Sections 5.5 & 5.6 Section 5.5: Greatest Common Factor and Factoring by Grouping Review: The difference between factors and terms Identify and factor out the Greatest Common Factor (GCF) Factoring

### Pre-Algebra 2. Unit 9. Polynomials Name Period

Pre-Algebra Unit 9 Polynomials Name Period 9.1A Add, Subtract, and Multiplying Polynomials (non-complex) Explain Add the following polynomials: 1) ( ) ( ) ) ( ) ( ) Subtract the following polynomials:

### To factor an expression means to write it as a product of factors instead of a sum of terms. The expression 3x

Factoring trinomials In general, we are factoring ax + bx + c where a, b, and c are real numbers. To factor an expression means to write it as a product of factors instead of a sum of terms. The expression

Adding and Subtracting Polynomials When you add polynomials, simply combine all like terms. When subtracting polynomials, do not forget to use parentheses when needed! Recall the distributive property:

### LESSON 6.2 POLYNOMIAL OPERATIONS I

LESSON 6. POLYNOMIAL OPERATIONS I LESSON 6. POLYNOMIALS OPERATIONS I 63 OVERVIEW Here's what you'll learn in this lesson: Adding and Subtracting a. Definition of polynomial, term, and coefficient b. Evaluating

### I CAN classify polynomials by degree and by the number of terms.

13-1 Polynomials I CAN classify polynomials by degree and by the number of terms. 13-1 Polynomials Insert Lesson Title Here Vocabulary monomial polynomial binomial trinomial degree of a polynomial 13-1

### LESSON 6.3 POLYNOMIAL OPERATIONS II

LESSON 6.3 POLYNOMIAL OPERATIONS II LESSON 6.3 POLYNOMIALS OPERATIONS II 277 OVERVIEW Here's what you'll learn in this lesson: Multiplying Binomials a. Multiplying binomials by the FOIL method b. Perfect

### MathB65 Ch 4 IV, V, VI.notebook. October 31, 2017

Part 4: Polynomials I. Exponents & Their Properties II. Negative Exponents III. Scientific Notation IV. Polynomials V. Addition & Subtraction of Polynomials VI. Multiplication of Polynomials VII. Greatest

### Lesson 3: Polynomials and Exponents, Part 1

Lesson 2: Introduction to Variables Assessment Lesson 3: Polynomials and Exponents, Part 1 When working with algebraic expressions, variables raised to a power play a major role. In this lesson, we look

### POLYNOMIAL EXPRESSIONS PART 1

POLYNOMIAL EXPRESSIONS PART 1 A polynomial is an expression that is a sum of one or more terms. Each term consists of one or more variables multiplied by a coefficient. Coefficients can be negative, so

### Math 101 Study Session Spring 2016 Test 4 Chapter 10, Chapter 11 Chapter 12 Section 1, and Chapter 12 Section 2

Math 101 Study Session Spring 2016 Test 4 Chapter 10, Chapter 11 Chapter 12 Section 1, and Chapter 12 Section 2 April 11, 2016 Chapter 10 Section 1: Addition and Subtraction of Polynomials A monomial is

### Unit 2-1: Factoring and Solving Quadratics. 0. I can add, subtract and multiply polynomial expressions

CP Algebra Unit -1: Factoring and Solving Quadratics NOTE PACKET Name: Period Learning Targets: 0. I can add, subtract and multiply polynomial expressions 1. I can factor using GCF.. I can factor by grouping.

### Review for Mastery. Integer Exponents. Zero Exponents Negative Exponents Negative Exponents in the Denominator. Definition.

LESSON 6- Review for Mastery Integer Exponents Remember that means 8. The base is, the exponent is positive. Exponents can also be 0 or negative. Zero Exponents Negative Exponents Negative Exponents in

### 5.1 Monomials. Algebra 2

. Monomials Algebra Goal : A..: Add, subtract, multiply, and simplify polynomials and rational expressions (e.g., multiply (x ) ( x + ); simplify 9x x. x Goal : Write numbers in scientific notation. Scientific

### 8-1 Factors and Greatest Common Factors 8-1. Factors and Greatest Common Factors

8-1 Factors and Greatest Common Factors Warm Up Lesson Presentation Lesson Quiz 1 2 pts 2 pts Bell Quiz 8-1 Tell whether the second number is a factor of the first number 1. 50, 6 2 pts no 2. 105, 7 3.

### Name: Chapter 7: Exponents and Polynomials

Name: Chapter 7: Exponents and Polynomials 7-1: Integer Exponents Objectives: Evaluate expressions containing zero and integer exponents. Simplify expressions containing zero and integer exponents. You

Solving Quadratic Equations MATH 101 College Algebra J. Robert Buchanan Department of Mathematics Summer 2012 Objectives In this lesson we will learn to: solve quadratic equations by factoring, solve quadratic

### Section 6.5 A General Factoring Strategy

Difference of Two Squares: a 2 b 2 = (a + b)(a b) NOTE: Sum of Two Squares, a 2 b 2, is not factorable Sum and Differences of Two Cubes: a 3 + b 3 = (a + b)(a 2 ab + b 2 ) a 3 b 3 = (a b)(a 2 + ab + b

### Chapter One: Pre-Geometry

Chapter One: Pre-Geometry Index: A: Solving Equations B: Factoring (GCF/DOTS) C: Factoring (Case Two leading into Case One) D: Factoring (Case One) E: Solving Quadratics F: Parallel and Perpendicular Lines

### Lesson 5b Solving Quadratic Equations

Lesson 5b Solving Quadratic Equations In this lesson, we will continue our work with Quadratics in this lesson and will learn several methods for solving quadratic equations. The first section will introduce

### LESSON EII.C EQUATIONS AND INEQUALITIES

LESSON EII.C EQUATIONS AND INEQUALITIES LESSON EII.C EQUATIONS AND INEQUALITIES 7 OVERVIEW Here s what you ll learn in this lesson: Linear a. Solving linear equations b. Solving linear inequalities Once

### Algebra I. Exponents and Polynomials. Name

Algebra I Exponents and Polynomials Name 1 2 UNIT SELF-TEST QUESTIONS The Unit Organizer #6 2 LAST UNIT /Experience NAME 4 BIGGER PICTURE DATE Operations with Numbers and Variables 1 CURRENT CURRENT UNIT

### Math 3 Variable Manipulation Part 3 Polynomials A

Math 3 Variable Manipulation Part 3 Polynomials A 1 MATH 1 & 2 REVIEW: VOCABULARY Constant: A term that does not have a variable is called a constant. Example: the number 5 is a constant because it does

### Math Lecture 18 Notes

Math 1010 - Lecture 18 Notes Dylan Zwick Fall 2009 In our last lecture we talked about how we can add, subtract, and multiply polynomials, and we figured out that, basically, if you can add, subtract,

### Read the following definitions and match them with the appropriate example(s) using the lines provided.

Algebraic Expressions Prepared by: Sa diyya Hendrickson Name: Date: Read the following definitions and match them with the appropriate example(s) using the lines provided. 1. Variable: A letter that is

### LESSON 8.1 RATIONAL EXPRESSIONS I

LESSON 8. RATIONAL EXPRESSIONS I LESSON 8. RATIONAL EXPRESSIONS I 7 OVERVIEW Here is what you'll learn in this lesson: Multiplying and Dividing a. Determining when a rational expression is undefined Almost

### Summer Prep Packet for students entering Algebra 2

Summer Prep Packet for students entering Algebra The following skills and concepts included in this packet are vital for your success in Algebra. The Mt. Hebron Math Department encourages all students

### Factoring Trinomials of the Form ax 2 + bx + c, a 1

Factoring Trinomials of the Form ax 2 + bx + c, a 1 When trinomials factor, the resulting terms are binomials. To help establish a procedure for solving these types of equations look at the following patterns.

### Order of Operations Practice: 1) =

Order of Operations Practice: 1) 24-12 3 + 6 = a) 6 b) 42 c) -6 d) 192 2) 36 + 3 3 (1/9) - 8 (12) = a) 130 b) 171 c) 183 d) 4,764 1 3) Evaluate: 12 2-4 2 ( - ½ ) + 2 (-3) 2 = 4) Evaluate 3y 2 + 8x =, when

### Real Numbers. Real numbers are divided into two types, rational numbers and irrational numbers

Real Numbers Real numbers are divided into two types, rational numbers and irrational numbers I. Rational Numbers: Any number that can be expressed as the quotient of two integers. (fraction). Any number

### P.1: Algebraic Expressions, Mathematical Models, and Real Numbers

Chapter P Prerequisites: Fundamental Concepts of Algebra Pre-calculus notes Date: P.1: Algebraic Expressions, Mathematical Models, and Real Numbers Algebraic expression: a combination of variables and

### A quadratic expression is a mathematical expression that can be written in the form 2

118 CHAPTER Algebra.6 FACTORING AND THE QUADRATIC EQUATION Textbook Reference Section 5. CLAST OBJECTIVES Factor a quadratic expression Find the roots of a quadratic equation A quadratic expression is

### Polynomials. This booklet belongs to: Period

HW Mark: 10 9 8 7 6 RE-Submit Polynomials This booklet belongs to: Period LESSON # DATE QUESTIONS FROM NOTES Questions that I find difficult Pg. Pg. Pg. Pg. Pg. Pg. Pg. Pg. Pg. Pg. REVIEW TEST Your teacher

### LESSON 13.1 NONLINEAR EQUATIONS

LESSON. NONLINEAR EQUATIONS LESSON. NONLINEAR EQUATIONS 58 OVERVIEW Here's what you'll learn in this lesson: Solving Equations a. Solving polynomial equations by factoring b. Solving quadratic type equations

### Day 131 Practice. What Can You Do With Polynomials?

Polynomials Monomial - a Number, a Variable or a PRODUCT of a number and a variable. Monomials cannot have radicals with variables inside, quotients of variables or variables with negative exponents. Degree

### Factor each expression. Remember, always find the GCF first. Then if applicable use the x-box method and also look for difference of squares.

NOTES 11: RATIONAL EXPRESSIONS AND EQUATIONS Name: Date: Period: Mrs. Nguyen s Initial: LESSON 11.1 SIMPLIFYING RATIONAL EXPRESSIONS Lesson Preview Review Factoring Skills and Simplifying Fractions Factor

### Welcome to Math Video Lessons. Stanley Ocken. Department of Mathematics The City College of New York Fall 2013

Welcome to Math 19500 Video Lessons Prof. Department of Mathematics The City College of New York Fall 2013 An important feature of the following Beamer slide presentations is that you, the reader, move

### Unit 2: Polynomials Guided Notes

Unit 2: Polynomials Guided Notes Name Period **If found, please return to Mrs. Brandley s room, M 8.** Self Assessment The following are the concepts you should know by the end of Unit 1. Periodically

### Unit 7: Factoring Quadratic Polynomials

Unit 7: Factoring Quadratic Polynomials A polynomial is represented by: where the coefficients are real numbers and the exponents are nonnegative integers. Side Note: Examples of real numbers: Examples

### Algebra I. Slide 1 / 216. Slide 2 / 216. Slide 3 / 216. Polynomials

Slide 1 / 216 Slide 2 / 216 lgebra I Polynomials 2015-11-02 www.njctl.org Table of ontents efinitions of Monomials, Polynomials and egrees dding and Subtracting Polynomials Multiplying a Polynomial by

### mn 3 17x 2 81y 4 z Algebra I Definitions of Monomials, Polynomials and Degrees 32,457 Slide 1 / 216 Slide 2 / 216 Slide 3 / 216 Slide 4 / 216

Slide 1 / 216 Slide 2 / 216 lgebra I Polynomials 2015-11-02 www.njctl.org Slide 3 / 216 Table of ontents efinitions of Monomials, Polynomials and egrees dding and Subtracting Polynomials Multiplying a

### Rising 8th Grade Math. Algebra 1 Summer Review Packet

Rising 8th Grade Math Algebra 1 Summer Review Packet 1. Clear parentheses using the distributive property. 2. Combine like terms within each side of the equal sign. Solving Multi-Step Equations 3. Add/subtract

### Lesson 3 Algebraic expression: - the result obtained by applying operations (+, -,, ) to a collection of numbers and/or variables o

Lesson 3 Algebraic expression: - the result obtained by applying operations (+, -,, ) to a collection of numbers and/or variables o o ( 1)(9) 3 ( 1) 3 9 1 Evaluate the second expression at the left, if

### Factoring Polynomials. Review and extend factoring skills. LEARN ABOUT the Math. Mai claims that, for any natural number n, the function

Factoring Polynomials GOAL Review and extend factoring skills. LEARN ABOUT the Math Mai claims that, for any natural number n, the function f (n) 5 n 3 1 3n 2 1 2n 1 6 always generates values that are

### Maintaining Mathematical Proficiency

Chapter 7 Maintaining Mathematical Proficiency Simplify the expression. 1. 5x 6 + 3x. 3t + 7 3t 4 3. 8s 4 + 4s 6 5s 4. 9m + 3 + m 3 + 5m 5. 4 3p 7 3p 4 1 z 1 + 4 6. ( ) 7. 6( x + ) 4 8. 3( h + 4) 3( h

### Polynomial comes from poly- (meaning "many") and -nomial (in this case meaning "term")... so it says "many terms

Polynomials Polynomial comes from poly- (meaning "many") and -nomial (in this case meaning "term")... so it says "many terms Polynomials A polynomial looks like this: Term A number, a variable, or the

### Study Guide for Math 095

Study Guide for Math 095 David G. Radcliffe November 7, 1994 1 The Real Number System Writing a fraction in lowest terms. 1. Find the largest number that will divide into both the numerator and the denominator.

### Polynomials and Polynomial Equations

Polynomials and Polynomial Equations A Polynomial is any expression that has constants, variables and exponents, and can be combined using addition, subtraction, multiplication and division, but: no division

### Geometry 21 Summer Work Packet Review and Study Guide

Geometry Summer Work Packet Review and Study Guide This study guide is designed to accompany the Geometry Summer Work Packet. Its purpose is to offer a review of the ten specific concepts covered in the

### Algebra 2/Trig Apps: Chapter 5 Quadratics Packet

Algebra /Trig Apps: Chapter 5 Quadratics Packet In this unit we will: Determine what the parameters a, h, and k do in the vertex form of a quadratic equation Determine the properties (vertex, axis of symmetry,

### 7-7 Multiplying Polynomials

Example 1: Multiplying Monomials A. (6y 3 )(3y 5 ) (6y 3 )(3y 5 ) (6 3)(y 3 y 5 ) 18y 8 Group factors with like bases together. B. (3mn 2 ) (9m 2 n) Example 1C: Multiplying Monomials Group factors with

### Algebra I Unit Report Summary

Algebra I Unit Report Summary No. Objective Code NCTM Standards Objective Title Real Numbers and Variables Unit - ( Ascend Default unit) 1. A01_01_01 H-A-B.1 Word Phrases As Algebraic Expressions 2. A01_01_02

### UNIT 9 (Chapter 7 BI) Polynomials and Factoring Name:

UNIT 9 (Chapter 7 BI) Polynomials and Factoring Name: The calendar and all assignments are subject to change. Students will be notified of any changes during class, so it is their responsibility to pay

### The number part of a term with a variable part. Terms that have the same variable parts. Constant terms are also like terms.

Algebra Notes Section 9.1: Add and Subtract Polynomials Objective(s): To be able to add and subtract polynomials. Recall: Coefficient (p. 97): Term of a polynomial (p. 97): Like Terms (p. 97): The number

### Unit 2: Polynomials Guided Notes

Unit 2: Polynomials Guided Notes Name Period **If found, please return to Mrs. Brandley s room, M 8.** Self Assessment The following are the concepts you should know by the end of Unit 1. Periodically

### Polynomials. In many problems, it is useful to write polynomials as products. For example, when solving equations: Example:

Polynomials Monomials: 10, 5x, 3x 2, x 3, 4x 2 y 6, or 5xyz 2. A monomial is a product of quantities some of which are unknown. Polynomials: 10 + 5x 3x 2 + x 3, or 4x 2 y 6 + 5xyz 2. A polynomial is a

### Divisibility Rules Algebra 9.0

Name Period Divisibility Rules Algebra 9.0 A Prime Number is a whole number whose only factors are 1 and itself. To find all of the prime numbers between 1 and 100, complete the following eercise: 1. Cross

### Unit 3A: Factoring & Solving Quadratic Equations After completion of this unit, you will be able to

Unit 3A: Factoring & Solving Quadratic Equations After completion of this unit, you will be able to Learning Target #1: Factoring Factor the GCF out of a polynomial Factor a polynomial when a = 1 Factor

### 20A. Build. Build and add. Build a rectangle and find the area (product). l e s s o n p r a c t i c e 1. X X X 2 + 6X X

l e s s o n p r a c t i c e 0A Build.. X X. X 6X 8 3. X 8 Build and add. 4. X 6X 3 3X 7X 9 5. X 8 X 6X 7 6. X 0X 7 X 8X 9 Build a rectangle and find the area (product). 7. (X )(X ) = 8. (X 4)(X 3) = 9.

### LESSON 10.1 QUADRATIC EQUATIONS I

LESSON 10.1 QUADRATIC EQUATIONS I LESSON 10.1 QUADRATIC EQUATIONS I 409 OVERVIEW Here s what you ll learn in this lesson: Solving by Factoring a. The standard form of a quadratic equation b. Putting a

### Intensive Math-Algebra I Mini-Lesson MA.912.A.4.3

Intensive Math-Algebra I Mini-Lesson M912.4.3 Summer 2013 Factoring Polynomials Student Packet Day 15 Name: Date: Benchmark M912.4.3 Factor polynomials expressions This benchmark will be assessed using

### LESSON 6.2 POLYNOMIAL OPERATIONS I

LESSON 6.2 POLYNOMIAL OPERATIONS I Overview In business, people use algebra everyday to find unknown quantities. For example, a manufacturer may use algebra to determine a product s selling price in order

### Solving Equations Quick Reference

Solving Equations Quick Reference Integer Rules Addition: If the signs are the same, add the numbers and keep the sign. If the signs are different, subtract the numbers and keep the sign of the number

### review To find the coefficient of all the terms in 15ab + 60bc 17ca: Coefficient of ab = 15 Coefficient of bc = 60 Coefficient of ca = -17

1. Revision Recall basic terms of algebraic expressions like Variable, Constant, Term, Coefficient, Polynomial etc. The coefficients of the terms in 4x 2 5xy + 6y 2 are Coefficient of 4x 2 is 4 Coefficient

### LESSON 6.1 EXPONENTS LESSON 6.1 EXPONENTS 253

LESSON 6.1 EXPONENTS LESSON 6.1 EXPONENTS 5 OVERVIEW Here's what you'll learn in this lesson: Properties of Exponents Definition of exponent, power, and base b. Multiplication Property c. Division Property

### Foundations of Math II Unit 5: Solving Equations

Foundations of Math II Unit 5: Solving Equations Academics High School Mathematics 5.1 Warm Up Solving Linear Equations Using Graphing, Tables, and Algebraic Properties On the graph below, graph the following

### Solving Linear Equations

Solving Linear Equations Golden Rule of Algebra: Do unto one side of the equal sign as you will do to the other Whatever you do on one side of the equal sign, you MUST do the same exact thing on the other

### Secondary Math 2H Unit 3 Notes: Factoring and Solving Quadratics

Secondary Math H Unit 3 Notes: Factoring and Solving Quadratics 3.1 Factoring out the Greatest Common Factor (GCF) Factoring: The reverse of multiplying. It means figuring out what you would multiply together

### SUMMER ASSIGNMENT FOR ALGEBRA II/TRIGONOMETRY

SUMMER ASSIGNMENT FOR ALGEBRA II/TRIGONOMETRY This summer assignment is designed to ensure that you are prepared for Algebra II/ Trigonometry. Nothing on this summer assignment is new. Everything is a

### Westside. Algebra 2 PreAP

Westside Algebra 2 PreAP Name Period IMPORTANT INSTRUCTIONS FOR STUDENTS!!! We understand that students come to Algebra II with different strengths and needs. For this reason, students have options for

### A monomial is measured by its degree To find its degree, we add up the exponents of all the variables of the monomial.

UNIT 6 POLYNOMIALS Polynomial (Definition) A monomial or a sum of monomials. A monomial is measured by its degree To find its degree, we add up the exponents of all the variables of the monomial. Ex. 2

### 8-1: Adding and Subtracting Polynomials

8-1: Adding and Subtracting Polynomials Objective: To classify, add, and subtract polynomials Warm Up: Simplify each expression. 1. x 3 7 x 9. 6(3x 4) 3. 7 ( x 8) 4 4. 5(4x (8x 6) monomial - A real number,

### Westside Algebra 2 PreAP

Westside Algebra 2 PreAP Name Period IMPORTANT INSTRUCTIONS FOR STUDENTS!!! We understand that students come to Algebra II with different strengths and needs. For this reason, students have options for