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.

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.

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 – f(3) = 45 – f(3) = 36

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

The Remainder Theorem 5x 2 – 4x – f(x) = 5x 2 – 4x + 3 f(3) = 5(3) 2 – 4(3) + 3 f(3) = 5 ∙ 9 – f(3) = 45 – 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 x) ∙ (x – 3) + 36

The Remainder Theorem Use synthetic substitution to find g(4) for the following function. f(x) = 5x 4 – 13x 3 – 14x 2 – 47x –13 –14 –

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

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 – –23 32 –

The Remainder Theorem Use synthetic substitution to find c(4) for the following function. c(x) = 2x 4 – 4x 3 – 7x 2 – 13x – –4 –7 –13 –

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

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

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.

The Factor Theorem x 3 + 4x 2 – 15x – 18 x – –15 – 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.

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

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.

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 x + 80 x – x 3 + 7x 2 – 33x – 18 x + 6 (x – 8)(x – 5)(x + 2) (x + 6)(2x + 1)(x – 3)

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

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)

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.

Factors of the constant Factors of the leading coefficient

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.

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

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