Remainder and Factor Theorems Unit 11

Definitions The real number, r, is a zero of f(x) iff:  r is a solution, or root, of f(x)=0  x-r is a factor of the expression that defines f (f(r)=0)  when the expression is divided by x- r, the remainder is 0  r is an x-intercept of the graph of f.

Remainder Theorem If the polynomial expression that defines the function of P is divided by x-r, then the remainder is the number P(r).

Factor Theorem x-r is a factor of the polynomial expression that defines the function P iff r is a solution of P(x)=0. That is, if P(r)=0.

Integer Roots Unit 11

Page 464, # ) 43) 44)

Warm Up Find the polynomial P(x) in the standard form that has roots of x = {-3, -1, 1} and P(0) = 9.

Quiz Factor & Remainder Theorem.

Example As the first step in creating a graph of the polynomial, find all x-intercepts of the polynomial :.

Rational Root Theorem Let P be a polynomial function with integer coefficients in standard form. If is a root of P(x) = 0, then  p is a factor of the constant term of P and  q is a factor of the leading coefficient of P.

Determine the number of roots. List all factors of the constant term. List all factors of the leading coefficient. List all the possible roots. Test each possible root to find the zeros of each polynomial.

Examples 1.) 2.)

Examples 3.) 4.)

Assignment Worksheet #1, 1-5

Show What You Know

Rational Roots Unit 11

Warm Up List all possible roots and use them to find the zeros of the polynomial:

Worksheet #1, 1-5 1) -6, -1, 1 2) -3, -2, 2 3) -4, 2 (multiplicity 2) 4) -1 (multiplicity 2), 2 5) -3, -1, 2, 3

Rational Root Theorem Let P be a polynomial function with integer coefficients in standard form. If is a root of P(x) = 0, then  p is a factor of the constant term of P and  q is a factor of the leading coefficient of P.

Determine the number of roots. List all factors of the constant term. List all factors of the leading coefficient. List all the possible roots. Test each possible root (using substitution or synthetic division) to find the zeros of each polynomial.

Examples 1.) 2.)

Assignment Worksheet #2, 1-16

Show What You Know

Rational Roots Unit 11

Warm Up List all possible roots and use them to find the zeros of the polynomial:

Worksheet #2, ) 1, 1/3 2) 1, 2, 4, 8, 16, 32, 64 3) 1, 2, 5, 10 4) 1, 2, 4, 8, 1/5, 2/5, 4/5, 8/5 5) 1, 5, 25, ½, 5/2, 25/2, ¼, 5/4, 25/4 6) 1, 3, 7, 21, 1/5, 3/5, 7/5, 21/5 7) 1, 3, 9, 27 8) 1, 7, ½, 7/2

Worksheet #2, ) x={1 (multiplicity 2), -3} 10) x={1 (multiplicity 2), 11} 11) x={-1 (multiplicity 2), -2} 12) x={-1, 1/5, -5} 13) x={1 (multiplicity 2), ¼} 14) x={-1, 1/3, -3} 15) x={1 (multiplicity 2), 1/5, 7} 16) x={-1 (multiplicity 2), 1/3, 5}

Quiz Integer and Rational Roots

Graphing Polynomials Unit 11

Warm Up Determine the number of roots. Then find the roots of the polynomial.

Critical Thinking In the process of solving you test 1, 2, 5, and 10 as possible zeros and determine that none of them are actual zeros. You then discover that -5/2 is a zero. You calculate the depressed polynomial to be Do you need to test 1, 2, 5, and 10 again? Why or why not?

End Behavior What happens to a polynomial function as its x- values get very small and very large is called the end behavior of the function.

End Behavior f(x)=ax n +… a > 0a < 0 leftrightleftright n is even n is odd

Leading Coefficient > 0

Leading Coefficient < 0

End Behavior f(x)=ax n +… a > 0a < 0 leftrightleftright n is even increasedecrease n is odd increasedecrease

Highest Exponent is Odd

Highest Exponent is Even

End Behavior f(x)=ax n +… a > 0a < 0 leftrightleftright n is even increase decrease n is odd decreaseincrease decrease

Examples Sketch the graph of each polynomial. 1.) 2.) 3.)

Assignment Worksheet 3, #1-8

Exit Survey Which of the following is the graph of ? A. B. C. D. B.

Polynomial Review Unit 11

Warm Up  Sketch a graph of the polynomial:

A=True B=False a) If f(-5)=0, then (x-5) is a factor of f(x). b) If x=9 is a root of f(x), then (x-9) is a factor of f(x). c) If the polynomial f(x) is synthetically divided by (x-4) and the remainder is 0, then f(4)=0. Example #1

Determine if (x+1) is a factor of the polynomial: A=Yes B=No Example #2

Example #3 Find the polynomial, in factored form, with the roots x={-2,2,4} and f(1)=18. A B C D

Example #4 How many roots will the function have? List all the possible rational roots. Perform the synthetic division. Write the polynomial in its factored form with each factor having only integer coefficients. Write the roots of the polynomial. Sketch the graph.

Assignment Review Sheet