EXAMPLE 3 Use synthetic division Divide f (x)= 2x 3 + x 2 – 8x + 5 by x + 3 using synthetic division. – 3 2 1 – 8 5 – 6 15 – 21 2 – 5 7 – 16 2x 3 + x 2.

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EXAMPLE 3 Use synthetic division

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EXAMPLE 3 Use synthetic division Divide f (x)= 2x 3 + x 2 – 8x + 5 by x + 3 using synthetic division. – – 8 5 – 6 15 – 21 2 – 5 7 – 16 2x 3 + x 2 – 8x + 5 x + 3 = 2x 2 – 5x + 7 – 16 x + 3 ANSWER SOLUTION

EXAMPLE 4 Factor a polynomial Factor f (x) = 3x 3 – 4x 2 – 28x – 16 completely given that x + 2 is a factor. SOLUTION Because x + 2 is a factor of f (x), you know that f (– 2) = 0. Use synthetic division to find the other factors. – 2 3 – 4 – 28 – 16 – – 10 – 8 0

EXAMPLE 4 Factor a polynomial Use the result to write f (x) as a product of two factors and then factor completely. f (x) = 3x 3 – 4x 2 – 28x – 16 Write original polynomial. = (x + 2)(3x 2 – 10x – 8 ) Write as a product of two factors. = (x + 2)(3x + 2)(x – 4) Factor trinomial.

GUIDED PRACTICE for Examples 3 and 4 Divide using synthetic division. 3. (x 3 + 4x 2 – x – 1)  (x + 3) SOLUTION (x 3 + 4x 2 – x – 1)  (x + 3) – – 1 – 1 – 3 – – 4 11 x x 2 – x – 1 x + 3 = x 2 + x – x + 3 ANSWER

GUIDED PRACTICE for Examples 3 and 4 4. (4x 3 + x 2 – 3x + 7)  (x – 1) SOLUTION (4x 3 + x 2 – 3x + 7)  (x – 1) – x 3 + x 2 – 3x + 1 x – 1 = 4x 2 + 5x x – 1 ANSWER

GUIDED PRACTICE for Examples 3 and 4 Factor the polynomial completely given that x – 4 is a factor. 5. f (x) = x 3 – 6x 2 + 5x + 12 SOLUTION Because x – 4 is a factor of f (x), you know that f (4) = 0. Use synthetic division to find the other factors. 4 1 – – 8 –12 1 – 2 – 3 0

GUIDED PRACTICE for Examples 3 and 4 Use the result to write f (x) as a product of two factors and then factor completely. f (x) = x 3 – 6x 2 + 5x + 12 Write original polynomial. = (x – 4)(x 2 – 2x – 3 ) Write as a product of two factors. = (x – 4)(x –3)(x + 1) Factor trinomial.

GUIDED PRACTICE for Examples 3 and 4 6. f (x) = x 3 – x 2 – 22x + 40 SOLUTION Because x – 4 is a factor of f (x), you know that f (4) = 0. Use synthetic division to find the other factors – – – 10 0

GUIDED PRACTICE for Examples 3 and 4 Use the result to write f (x) as a product of two factors and then factor completely. f (x) = x 3 – x 2 – 22x + 40 Write original polynomial. = (x – 4)(x 2 + 3x – 10 ) Write as a product of two factors. = (x – 4)(x –2)(x +5) Factor trinomial.