2.5 Apply the Remainder and Factor Theorem

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Presentation transcript:

2.5 Apply the Remainder and Factor Theorem Pg. 85

Factor Theorem (x – r) is a “factor” of the polynomial expression that defines the function P if and only if…….. “r” is a solution of P(x) = 0, that is, P(r) = 0.

Factor Theorem Use substitution to determine whether (x + 2) is a factor of x3 – 2x2 – 5x + 6 If (x + 2) is a factor then x = - 2 (-2)3 – 2(-2)2 – 5(-2) + 6 = (-8) – 8 + 10 + 6 = = 0 … yes (x + 2) is a factor

Dividing Polynomials Dividing Polynomials to get the factors can be done in one of two ways Long Division or Synthetic Division Lets start with Long Division

Long Division Now you must ask yourself, “what can I multiply “x” by to get “x3” The answer is = x2 Binomial Divisor – You must start with two terms in the dividend.

Factoring by Long Division Answer Distribute Negative and Add Remainder if there is any

Polynomial Long Division All degrees of the polynomial must be represented, if not then placeholders must be used Use zero (0) for each degree not represented Divide by

Factoring by Synthetic Division (only with a Binomial) (x-2) is a Factor…x=2 of Drop 1st coefficient down Use the Polynomials coefficients Remainder if there is any Coefficients of the quotient

Synthetic Division All degrees of the polynomial must be represented, if not then placeholders must be used Use zero (0) for each degree not represented Since the divisor is a binomial we can use Synthetic Division Divide by

Factoring a Polynomial Factor given that (x – 3) is a factor

Finding Zeros of a Function Find the other zeros of given that f(-1) = 0. Remember zeros are solutions or roots (x = )

Example (divide)

Homework Pg. 87, 2 – 24 (even), 25 Pg. 88, 2 – 16 (even)