Pre-Algebra 2. Unit 9. Polynomials Name Period

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1 Pre-Algebra Unit 9 Polynomials Name Period

3 9.1A Add, Subtract, and Multiplying Polynomials (non-complex) Explain Add the following polynomials: 1) ( ) ( ) ) ( ) ( ) Subtract the following polynomials: 1) ( ) ( ) ) ( ) ( ) FOIL Method for Multiplying Binomials FOIL is an acronym for First, Outer, Inner, Last. Multiply these terms together and then find their sum. You will notice that the box method and FOIL represent the same data. Example: O x 3 x 5 I Place the letters F, O, I, and L in the boxes to match the same multiplication as in the FOIL example. x x + 3 F L -5 Total Area: Total Area: 1) Use FOIL to multiply the following polynomials. 3 x x 5 x Total Area: ) Now, use the box method to multiply the polynomials x 3 x x 5 1 Total Area:

4 3) What does (x) look like when it s written in expanded form? 4) What does (x + 3) written in expanded form? 5) Draw your own box to find the simplified form of (x + 3). Multiply the following polynomials: 1) ( )( )( ) ) ( )( )( ) 3) ( )( )( ) Extra: One factor of ( ) is ( ). What are the other two factors? Area Problems 1. Find the area.. Find the area x y 5 3 4x y 6x 3 y 19x 3 y. Find the width. 6. Find the length.? A x y 7 9xy x y A 5x? 3 y

5 9.1A Add, Subtract, and Multiply Polynomials WS Simplify the following polynomials 1) ( ) ( ) ) ( ) ( ) 3) ( ) ( ) Use the FOIL method to find the area of a rectangle with the following dimensions: 4) (x 4)(3x 7) 5) (x 4y)(-x -5y) 6) (3x + 5)(x + 4) Use the box method to simplify the following: 7) (x)(-4x + 8) 8) (3x 4) 9) (3x + 4)(3x 4) Use the box to find the missing amounts: 10) ( )( ) ) 11) ( )( ) ) Use a box to determine the dimensions that would create the following area: 1) X + 5x 6 13) 5x + 40x

6 1. Please find the area.. Please find the area. 7x 3 y 8 z 9 x 15x y 6 z y z 3. Please find the length. 4. Please find the width x yz 9 57y z 6 3 A 105x y z 9 3? A 19y z? 4

7 9.1B Add, Subtract, and Multiplying Complex Numbers Explain Add the following polynomials: 1) ( ) ( ) ( ) ) ( ) ( ) ( ) Subtract the following polynomials: 1) ( ) ( ) ) ( ) ( ) ( ) Multiply the following polynomials: 1) ( )( ) ) ( )( ) 3) ( )( ) 5

8 9.1B Add, Subtract, and Multiplying Complex Numbers WS Add the following polynomials: 1) ( ) ( ) ( ) ) ( ) ( ) Subtract the following polynomials: 1) ( ) ( ) ) ( ) ( ) ( ) Multiply the following polynomials: 1) ( )( ) ) ( )( ) 3) ( )( ) 6

9 9. Notes: Long Division of Polynomials Name Algebra II Date Per Long Division Steps: 1. Divide the first term of the dividend by the first term of the divisor.. Write the result from step 1 in the quotient and use it to multiply the divisor. 3. Subtract the product from the dividend. 4. Repeat steps 1-3 using the difference from step 3 as the new dividend. Example: Find the quotient Example: Find the quotient Example: Find the quotient. x 3 3x 3x x x 1 3 Dividend = x 3x 3x Divisor = x x 1 x x x x x x 3 x x x x x x x 0 remainder 7

10 Use long division. 1. ( x 5x 14) ( x ). ( x x 48) ( x 6) 3. 3 ( x 3x 16x 1) ( x 1) 4. 3 ( x 3x 8x 5) ( x 1) 5. 4 ( x 7x 9x 10) ( x ) 6. 3 ( x x 1x 45) ( x x 15) 7. 3 (8x 5x 1x 10) ( x 3) (4x x 9x 1) ( x x) 8

11 9. WS: Long Division of Polynomials Name Algebra II Date Per Use long division. 1. ( x 6x 8) ( x 4). (x 7x 10) ( x ) 3. 3 ( x 10x 19x 30) ( x 6) 4. 3 ( x 4x 6) ( x 3) 5. 4 (4x 15x 4) ( x ) 6. 3 (3x 11x 4x 1) ( x x) 9

12 ( x 5x 8x 13x 1) ( x 6) 8. 3 (3x 34x 7x 64) (3x ) (7x 11x 7x 5) ( x 1) 10. (x 3x x x 4) ( x x 1) 10

13 9.3 Notes: Synthetic Division of Polynomials Name Algebra II Date Per Synthetic Division Steps: 1. Write the coefficients of the polynomial and then write the value of r on the left. Write the first coefficient below the line.. Multiply the r-value by the number below the line, and write the product below the next coefficient. 3. Write the sum (not the difference) below the line. Multiply r by the number below the line and write the product below the next coefficient. 4. Write the sum (not the difference) below the line. Repeat steps 1-3 as needed. Note: Synthetic Division can only be used on linear divisors (i.e. in the form x - r). If the divisor is in any other form, Long Division must be used. Use synthetic division. 1. ( x 5x 6) ( x 1). 3 ( x x 30) ( x 3) (x 11x 15x 6x 18) ( x 3) 4. ( x x 48) ( x 5) 5. 4 ( x 7x 9x 10) ( x ) 6. 3 (3x 16x 103x 36) ( x 4) 11

14 Synthetic division can be used to factor. In example, x 3 + x + 30 was divided by x + 3 and the answer was. Thus, x 3 + x + 30 = ( )( ) 7. Factor f(x) = x 3 18x + 95x 16 given that x 9 is a factor. 8. Factor f(x) = x 3 + 3x 39x 0 given that -5 is a zero of f(x). Remainder Theorem Remainder & Factor Theorems If a polynomial f(x) is divided by x a, the remainder is the constant f(a), and: f(x) = q(x) (x a) + f(a) where q(x) is a polynomial with degree one less that the degree of f(x). qx ( ) x a f ( x) WORK f( a) 9. Let f(x) = x 4 x Show that f(-1) is the remainder when f(x) is divided by x Let f(x) = x 3 + 5x 7x +. Find f(). Factor Theorem Let a polynomial f(x) be divided by x a. If the remainder is 0, then x a is a factor of f(x). 11. Let f(x) = 3x 3 4x 8x 16. Is x + a factor? 1

15 9.3 WS: Synthetic Division of Polynomials Name Algebra II Date Per Use synthetic division. 1. ( x 7x 1) ( x 3). (4x 13x 5) ( x ) 3. 3 ( x 4x 6) ( x 3) 4. 3 ( x 5x ) ( x 4) 5. 3 ( x 6x 5x 1) ( x 4) 6. 3 ( x 18x 95x 150) ( x 10) ( x 5x 8x 13x 1) ( x 6) 8. 3 (4x 7x 3x 64) ( x 7) 13

16 ( x 4x 13x 4x 1) ( x 1) 10. ( x 6x 40x 33) ( x 7) Use the Remainder Theorem to determine the following. 11. Let f(x) = 3x 4 x 3 + x 3. Find f(-1). 1. Let f(x) = x 3 + x 5x. Find f(3). Use the Factor Theorem to determine the following. 13. Let f(x) = x 4 4x 3 0x + 48x. Is (x + 4) a factor? 14. Let f(x) = x 3 x 5x + 3. Is (x 3) a factor? 14

17 9.4 Solving by Factoring (grouping) Elaborate Activity 1. Write the standard form for a quadratic trinomial:. Given the trinomial, x 5x 1, we will express this as the product of two binomials. Identify A Identify B Identify C Our job will be easier if we can rewrite this trinomial as a polynomial with 4 terms. Then we can group the terms in pairs and use GCF factoring!!!!! 3. When we multiply the coefficients of the first and last terms, AC, we have our target product. x 5x 1 times -1 equals -4 AC, our target product =. Our target sum, the coefficient of the middle term, B, =. We need to find factors of the target product, -4, that add to equal the target sum, -5, in order to begin factoring our trinomial. 4. Using, x 5x 1, find the pair of factors for AC, our target product that we need for our target sum, -5. Since AC = -4, let s check some factors and their sums in the following chart: Factors of -4 Sum Highlight the one which has the target product of -4 and target sum of Page 1 of 4

18 5. Let s rewrite x 5x 1 as a polynomial with 4 terms, using our target factors of 8 and 3. x 5x 1 Highlight what changed. x 8x + 3x 1 We have only changed the form of the original trinomial but the polynomial is the same. 6. x 5x 1 x 8x + 3x 1 By grouping (circle the first two and second two together) these 4 terms in pairs, we can factor out the GCF from each pair. x 8x + 3x 1 (x 4) + (x 4) Write these two terms as one binomial factor, (x + 3) and the binomial (x 4) is our other factor. So, x 5x 1 in factored form, is (x + 3)(x 4). Check by multiplying (x + 3)(x 4) to see if the result is x 5x Let s factor another trinomial. Factor 4x 9x + 5, using the AC method we just learned. AC, our target product =. Our target sum =. We need to find factors of the target product, that add to equal the target sum,. Factors of 0 Addition of Factors Sum of Factors Result? Rewrite 4x 9x + 5 as a polynomial with 4 terms, using our target factors of and. So, 4x 9x + 5 in factored form, is ( )( ). 8. Factor 3x + x + 7. target product = target sum = 16 Page of 4

19 Evaluate (Assignment) Factor completely. 1. y + 7y + 5. r 0r x + 7x a + 10a 8 5. d + 4d 1 6. p + 14p x + 11x q q 17 Page 3 of 4

20 9. 5x 3x w 3 + w 35w 11. 4h + 8h 4 1. k + 1k x + 7x x + 13x y 3 8y 4y 18 Page 4 of 4

21 9.5 Notes: Polynomial Behavior PAP Algebra II Name Date Per Function Factored Form Graph of f(x) Root (x = ) Crosses, Tangent, Wiggles & Power of Factor End Behavior x = - Left Right Term of Highest Degree & Odd or Even ax n 1. f( x) ( x )( x 1)( x 3). f x x x x ( ) ( )( ) ( 6) 3. f( x) ( x )( x 5) ( x 3) 4. f x x x x 3 ( ) ( )( 1) ( 1) 5. f x x x x 3 ( ) ( ) ( 1) ( 4) 1. Graph each function on your graphing calculator.. Sketch the graph on the chart. (Do not worry about scale. We are just interested in x-intercepts and end behavior.) 3. Fill in the remaining columns of the chart based on your graph. 4. What does the power of a factor tell you about how f(x) crosses the x-axis at the corresponding root? n 5. If f(x) has a highest powered term of ax, how does its end behavior look if: a > 0, n even: a > 0, n odd: a < 0, n even: a < 0, n odd: 19

22 9.5 WS: Polynomial Behavior PAP Algebra II Name Date Per Sketch the graph on the chart. (Do not worry about scale. We are just interested in x-intercepts and end behavior.) Fill in the remaining columns of the chart based on your graph. Function Graph of f(x) Crosses, Tangent, Wiggles & End Behavior Power of Factor x = - x = 1 x = 3 Left Right Term of Highest Degree ax n 1. f( x) ( x ) ( x 1)( x 3) Wiggles 3 Crosses 1 Tangent Up Up 1 x 6. f( x) ( x ) ( x 1)( x 3) 3. f( x) ( x )( x 1) ( x 3) 3 4. f( x) ( x )( x 1) ( x 3) 4 5. f x x x x ( ) ( ) ( 1)( 3) 6. f x x x x 5 ( ) ( ) ( 1)( 3) 7. f x x x x ( ) ( )( 1) ( 3) 8. f ( x) ( x 1) 3 0

Polynomials. Exponents. End Behavior. Writing. Solving Factoring. Graphing. End Behavior. Polynomial Notes. Synthetic Division.

Polynomials. Exponents. End Behavior. Writing. Solving Factoring. Graphing. End Behavior. Polynomial Notes. Synthetic Division. Polynomials Polynomials 1. P 1: Exponents 2. P 2: Factoring Polynomials 3. P 3: End Behavior 4. P 4: Fundamental Theorem of Algebra Writing real root x= 10 or (x+10) local maximum Exponents real root x=10