c Is 5 x a factor of 3 2 19 10 p xx xx d Is 1 x a factor of 32 7 11 3 p x x x x

C is 5 x a factor of 3 2 19 10 p xx xx d is 1 x a

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(c) Is 5 x a factor of 3 2 19 10 p x x x x ? (d) Is 1 x a factor of 3 2 7 11 3 p x x x x     2 2 11 22 9 9 11 9 22 4 p x x x p (4) For x a to be a factor of p x , the remainder, or p a , must be equal to zero. This is exactly what it means for one integers to be a factor of another integer, i.e. upon division the remainder is zero. 2 3 3 11 3 24 0 Yes, 3 is a factor p x 2 5 2 5 9 5 2 3 No, 5 is not a factor. p x 3 2 1 1 7 1 11 1 3 0 Yes, 1 is a factor p x         3 2 5 5 5 19 5 10 5 No, 5 is not a factor p x   If 4 x is a factor of 2 52 x kx then 4 x must be a zero of the same expression, i.e:     2 4 4 52 0 16 4 52 0 4 36 0 9 k k k k
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Name: ____________________________________ Date: __________________ C OMMON C ORE A LGEBRA II, U NIT #10 P OLYNOMIAL AND R ATIONAL F UNCTIONS L ESSON #11 e M ATH I NSTRUCTION , R ED H OOK , NY 12571, © 2015 T HE R EMAINDER T HEOREM C OMMON C ORE A LGEBRA II H OMEWORK F LUENCY 1. Which of the following is the remainder when the polynomial 2 5 3 x x is divided by the binomial 8 x ? (1) 107 (3) 3 (2) 27 (4) 9 2. If the ratio 2 2 17 42 5 x x x is placed in the form 5 r q x x , where q x is a polynomial, then which of the following is the correct value of r ? (1) 3 (3) 18 (2) 177 (4) 7 3. When the polynomial p x was divided by the factor 7 x the result was 11 7 x x . Which of the following is the value of 7 p ? (1) 8 (3) 11 (2) 7 (4) It does not exist 4. Which of the following binomials is a factor of the quadratic 2 4 35 24 x x ? Try to do this without factoring but by using the Remainder Theorem. (1) 6 x (3) 8 x (2) 4 x (4) 2 x 5. Which of the following linear expressions is a factor of the cubic polynomial 3 2 9 16 12 x x x ? (1) 6 x (3) 3 x (2) 1 x (4) 2 x Answer Key     2 2 5 3 8 8 5 8 3 27 p x x x p (2) 2 2 2 17 42 5 2 5 17 5 42 7 p x x x r p r (4) 7 11 7 11 7 7 7 p x p q x x p x x x (3) Test the zeroes of each linear factor to see if they are zeroes of the polynomial. Of our choices, only 8 x is a zero, thus 8 x is a factor. (3) This is the same as #4. Check the zeroes of each linear expression to see which is a zero of the cubic. Only 6 x   is a zero, thus 6 x is a factor. (1)
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C OMMON C ORE A LGEBRA II, U NIT #10 P OLYNOMIAL AND R ATIONAL F UNCTIONS L ESSON #11 e M ATH I NSTRUCTION , R ED H OOK , NY 12571, © 2015 6. Consider the cubic polynomial 3 2 46 80 p x x x x . (a) Using polynomial long division, write the ratio of 3 p x x in quotient-remainder form , i.e. in the form 3 r q x x . Evaluate   3 p . How does this help you check your quotient-remainder form? (b) Evaluate   5 p . What does this tell you about the binomial