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The Remainder and Factor Theorems Check for Understanding 2.3 – Factor polynomials using a variety of methods including the factor theorem, synthetic division, long division, sums and differences of cubes, and grouping.

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The Remainder Theorem 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 than the degree of f(x). Dividend equals quotient times divisor plus remainder.

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The Remainder Theorem Find f(3) for the following polynomial function. f(x) = 5x 2 – 4x + 3 f(3) = 5(3) 2 – 4(3) + 3 f(3) = 5 ∙ 9 – 12 + 3 f(3) = 45 – 12 + 3 f(3) = 36

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The Remainder Theorem Now divide the same polynomial by (x – 3). 5x 2 – 4x + 3 3 5 –4 3 53611 33 15

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The Remainder Theorem 5x 2 – 4x + 3 3 5 –4 3 15 33 5 11 36 f(x) = 5x 2 – 4x + 3 f(3) = 5(3) 2 – 4(3) + 3 f(3) = 5 ∙ 9 – 12 + 3 f(3) = 45 – 12 + 3 f(3) = 36 Notice that the value obtained when evaluating the function at f(3) and the value of the remainder when dividing the polynomial by x – 3 are the same. Dividend equals quotient times divisor plus remainder. 5x 2 – 4x + 3 = (5x 2 + 11x) ∙ (x – 3) + 36

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The Remainder Theorem Use synthetic substitution to find g(4) for the following function. f(x) = 5x 4 – 13x 3 – 14x 2 – 47x + 1 4 5 –13 –14 –47 1 20 28 56 36 5 7 14 9 37

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The Remainder Theorem Synthetic Substitution – using synthetic division to evaluate a function This is especially helpful for polynomials with degree greater than 2.

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The Remainder Theorem Use synthetic substitution to find g(–2) for the following function. f(x) = 5x 4 – 13x 3 – 14x 2 – 47x + 1 –2 5 –13 –14 –47 1 –10 46 –64 222 5 –23 32 –111 223

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The Remainder Theorem Use synthetic substitution to find c(4) for the following function. c(x) = 2x 4 – 4x 3 – 7x 2 – 13x – 10 4 2 –4 –7 –13 –10 8 16 36 92 2 4 9 23 82

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Time for Class work Time for Class work Evaluate each function at the given value.

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The Factor Theorem The binomial (x – a) is a factor of the polynomial f(x) if and only if f(a) = 0.

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The Factor Theorem When a polynomial is divided by one of its binomial factors, the quotient is called a depressed polynomial. If the remainder (last number in a depressed polynomial) is zero, that means f(#) = 0. This also means that the divisor resulting in a remainder of zero is a factor of the polynomial.

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The Factor Theorem x 3 + 4x 2 – 15x – 18 x – 3 3 1 4 –15 –18 3 21 18 1 7 6 0 Since the remainder is zero, (x – 3) is a factor of x 3 + 4x 2 – 15x – 18. This also allows us to find the remaining factors of the polynomial by factoring the depressed polynomial.

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The Factor Theorem x 3 + 4x 2 – 15x – 18 x – 3 3 1 4 –15 –18 3 21 18 1 7 6 0 x 2 + 7x + 6 (x + 6)(x + 1) The factors of x 3 + 4x 2 – 15x – 18 are (x – 3)(x + 6)(x + 1).

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The Factor Theorem (x – 3)(x + 6)(x + 1). Compare the factors of the polynomials to the zeros as seen on the graph of x 3 + 4x 2 – 15x – 18.

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The Factor Theorem Given a polynomial and one of its factors, find the remaining factors of the polynomial. Some factors may not be binomials. 1.x 3 – 11x 2 + 14x + 80 x – 8 2. 2x 3 + 7x 2 – 33x – 18 x + 6 (x – 8)(x – 5)(x + 2) (x + 6)(2x + 1)(x – 3)

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Using the Factor Theorem, determine if f(x) is a factor of p(x) The Factor Theorem

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Time for Class work Time for Class work Using the Factor Theorem, factor fully each of the following polynomials: Using the Factor Theorem, determine if f(x) is a factor of p(x)

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The Rational Zero Theorem The Rational Zero Theorem gives a list of possible rational zeros of a polynomial function. Equivalently, the theorem gives all possible rational roots of a polynomial equation. Not every number in the list will be a zero of the function, but every rational zero of the polynomial function will appear somewhere in the list. The Rational Zero Theorem If f (x) a n x n a n-1 x n-1 … a 1 x a 0 has integer coefficients and (where is reduced) is a rational zero, then p is a factor of the constant term a 0 and q is a factor of the leading coefficient a n. The Rational Zero Theorem If f (x) a n x n a n-1 x n-1 … a 1 x a 0 has integer coefficients and (where is reduced) is a rational zero, then p is a factor of the constant term a 0 and q is a factor of the leading coefficient a n.

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Factors of the constant Factors of the leading coefficient

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EXAMPLE : Using the Rational Zero Theorem List all possible rational zeros of f (x) 15x 3 14x 2 3x – 2. Solution The constant term is 2 and the leading coefficient is 15. Divide 1 and 2 by 1. Divide 1 and 2 by 3. Divide 1 and 2 by 5. Divide 1 and 2 by 15. There are 16 possible rational zeros. The actual solution set to f (x) 15x 3 14x 2 3x – 2 = 0 is {-1, 1 / 3, 2 / 5 }, which contains 3 of the 16 possible solutions.

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22 The Rational Zero Test Example Find all potential rational zeros of Solution

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23 The Rational Zero Test (continued) Example Use the Rational Zero Test to find ALL rational zeros of

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