6.5 The Remainder and Factor Theorems

Goals p Divide polynomials and relate the result to the remainder theorem and the factor theorem. p Use polynomial in real-life problems.

Your Notes VOCABULARY

Polynomial A method used to divide polynomials similar to the way you divide numbers

Synthetic division A method used to divide a polynomial by an expression of the form x k

Example 1 Using

.Divide 4x4 ؊ x2 ؊ 18x ؉ 8 by x2 ؉ 2x ؉ 3 Write division in the same format you would use when dividing numbers. Include a “0” as the coefficient of x3. 4 3 2 4x 8x 3x x2 x2 x2

4x2 8x 3 x2 2x 3ͤ4ෆxෆ4ෆෆ0xෆ3ෆෆ1ෆ2ෆxෆ2ෆෆෆ18ෆxෆෆෆ8 4x4 8x3 12x2

8x3 13x2 18x 8x3 16x2 24x

3x2 6x 8 3x2 6x 9

1 Write the result as follows. 4x4 x2 18x 8 1  4x2 8x 3  x2 2x 3 x2 2x 3

136 2 Notetaking Guide • Chapter 6 Your Notes REMAINDER THEOREM If a polynomial f(x) is divided by x k, the remainder is r f(k) .

Example 2 Using

Divide x3 x2 5x 3 by x 2.

Solution To find the value of k, rewrite the divisor in the form x k. Because x 2 x (2) , k 2 . 2 1153 226 1 1 39 x3 x2 5x 3 9 x2 x 3  x 2 x 2

FACTOR THEOREM A polynomial f(x) has a factor x k if and only if f(k) 0 .

Example 3 Factoring a Polynomial

Factor f(x) x3 19x 30 given that f(5) 0.

Solution Because f(5) 0, you know that x 5 is a factor of f(x). Use synthetic division to find the other factors. 5 1019 30 52530 15 6 0 The result gives the coefficients of the . x3 19x 30 ( x 5 )( x2 5x 6 ) ( x 5 )( x 2 )( x 3 )

Lesson 6.5 • Algebra 2 Notetaking Guide 137 Your Notes Example 4 Finding Zeros of a Polynomial

A zero of f(x) x3 x2 4x 4 is x 1. Find the other zeros.

Solution Because f(1) 0, you know that x 1 is a factor of f(x). Use synthetic division to find the other factors. 1 114 4 104 1040 The result gives the coefficients of the quotient. f(x) x3 x2 4x 4 ( x 1 )( x2 4 ) ( x 1 )( x 2 )( x 2 ) By the factor theorem, the zeros of f are 1, 2, and 2 .

Checkpoint Complete the following exercises.

1. Use long division to divide x2 4x 1 by x 3.

4 x 1  x 3

2. Use synthetic division to divide 2x3 x2 3x 4 by x 1.

2 2x2 3x 6  x 1

3. Factor f(x) 2x3 x2 25x 12 given that f(4) 0.

Homework (x 4)(x 3)(2x 1) 4. A zero of f(x) x4 5x2 4 is 1. Find the other zeros.

1, 2, 2

138 Algebra 2 Notetaking Guide • Chapter 6