8-1: Adding and Subtracting Polynomials
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1 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) (4x (8x 6) monomial - A real number, a variable, or a product of a real number and one or more variables with 7 whole number exponents. (For example: 16, z, 3c,.5xy ) degree of a monomial - The sum of the exponents of its variables. Example 1: What is the degree of each monomial? 4 3 a) 7d b) x y c) 18 d) 5 9m n Example : Simplify each expression by adding or subtracting like terms a) 6g g b) 5a c 7a c polynomial - A monomial or a sum of monomials. (For example: 6 4 3x x 8x standard form - The degrees of the monomial terms decrease from left to right. degree of a polynomial - The highest degree of any monomial term of the polynomial. Example 3: 3 Consider the polynomial 7x 5x 4x. a) Write the expression in standard form. b) What is the degree of the polynomial? Classifying Polynomials by Degree: Classifying Polynomials by Number of Terms: Degree Name # of Terms Name 0 constant 1 monomial 1 linear binomial quadratic 3 trinomial 3 cubic
2 Example 4: Write each expression in standard form. Then name each polynomial by its degree and number of terms. a) 8c 4c 3 b) 5w 3 w c) 7 d) 9 8h e) y 6y y 1 Example 5: Simplify each sum. a) (8a 3a 9) (5a 7a 4) b) 4 4 (3 x 6 x ) ( x x 11) Example 6: Simplify each difference. a) (3c 8c (7c c 4) b) 5 5 ( 6z 4z 3) (z 9z 3z Example 7: A nutritionist studied the U.S. consumption of carrots and broccoli over a 6-year period. The nutritionist modeled the results, in millions of pounds, with the following polynomials. 3 Carrots: C( x) 9x 15x 6x 4477 Broccoli: B( x) 18x 14x 1535 In each polynomial, x = 0 corresponds to the first year in the 6-year period. What polynomial models the total number of pounds, in millions, of carrots and broccoli consumed in the United States during the 6-year period? Closure Question: How are the processes of adding and subtracting polynomials different?
3 8-: Multiplying and Factoring Objective: To multiply a polynomial by a monomial To factor a monomial from a polynomial Warm Up: Simplify each expression. Write each answer in standard form. 1. (5x 6x 7) (x 9x. (3w 5w ) (8w w 6) 3. 8( c 4. 7(5 h) 10(4h 3) Find the greatest common factor. 5. 9, 15, , 0, 8 Example 1: Simplify each product. Write each answer in standard form. 3 a) g(3g 7g 5) b) 3 d ( 4d 6d 8d Example : Simplify each expression. Write each answer in standard form. a) y( y ) 3 y( y 5) b) a ( a a( a 3) greatest common factor (GCF) - The greatest factor that divides evenly into each term of an expression. Factoring a polynomial reverses the multiplication process. When factoring a monomial from a polynomial, the first step is to find the GCF of the polynomial s terms.
4 Example 3: Find the GCF of the terms. 4 a) x 10x 6x b) c 10c 5c Example 4: Factor each polynomial. a) 16n 1n 4 b) 9z 15z c) e 1e 7e Example 5: A circular hedge surrounds a sculpture on a square base. The radius of the hedge is 6x. The side length of the square sculpture base is 4x. What is the area of the hedge? Write your answer in factored form and leave in terms of. Closure Question: Explain how to find the GCF of a polynomial.
5 8-3 & 8-4: Multiplying Binomials Objective: To multiply two binomials and a binomial by a trinomial Warm Up: Simplify each expression. Write each answer in standard form n( n 8n 3). y (y 7) 3. w( w 4 w( w 7) 4. 6 c( c ) c(8c 3) When multiplying two polynomials, you use the extended distributive property by multiplying each term of the first polynomial by each term of the second polynomial. You can use the box method to help organize your work. Using this method, you will be able to see the polynomials as the dimensions and the product as the area of each box. Example 1: Simplify each product. Write each answer in standard form. a) ( x 3)( x 9) b) (6a )(5a Difference of Squares Pattern and Square of Binomial Pattern: Some pairs of binomials have special products. If you learn to recognize these pairs, finding the product of two binomials will sometimes be quicker and easier. Example : Simplify each product. Write each answer in standard form. a) (4 z 5)(4 z 5) b) (7d 3) Example 3: Simplify each product. Write each answer in standard form. a) ( h 6)( h ) b) (4e 1)( e 8)
6 Example 4: Simplify each product. Write each answer in standard form. a) ( y 3)( y 5y 9) b) (4m m 5)(6m 3) Example 5: A community center is expanding the size of its rectangular meeting room. The room is currently 100 feet long and 70 feet wide. The center plans to expand both the length and the width of the meeting room by x feet. What polynomial in standard form represents the area of the expanded meeting room? Example 6: Find the area of the shaded region. Write each answer in standard form. a) b) Closure Question: When multiplying two binomials, what do the binomials represent and what does the product represent?
7 Objective: To factor trinomials of the type 8-5 & 8-7: Factoring x +bx+c ax bx c (where a = 1) Warm Up: Simplify each product. 1. ( c )( c 5). ( a 4)( a 6) 3. ( n 7)( n 7) 4. ( r 3) Look at the warm-up problems to show the idea of factoring. Do you see a pattern? Example 1: Factor x 8x 15 Example : Factor e 16e 48 Example 3: Factor y 6y 7
8 Example 4: Factor d 8d 9 Example 5: Factor g 5 Example 6: Factor m 0m 100 Example 7: The area of a rectangular picture frame is given by the trinomial possible dimensions of the picture frame? x 6x 16. What are the Closure Question: How can you determine what numbers are used in the binomial factors when factoring expressions of the type ax bx c?
9 Objective: To factor trinomials of the type 8-6 & 8-7: Factoring ax +bx+c ax bx c (where a Warm Up: Factor out the GCF. 1. 5x 10x a 3a 8a Factor each expression. 3. y 11y 4 4. c 6c 40 Before factoring with the box, first remember to check if a GCF can be factored out of the polynomial. Example 1: Factor 3x 5x Example : Factor k 1k 11 Example 3: Factor 5w 6w 8
10 Example 4: Factor 5a 9 Example 5: Factor 4c 8c 49 Example 6: Factor 8y y 14 Example 7: Factor 18d 1d 6 Example 8: The area of a rectangular rug is given by the trinomial dimensions of the rug? 6n 19n 15. What are the possible Closure Question: What is the first thing you should look for when factoring a trinomial?
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