EXAMPLE 2 Find all zeros of f (x) = x 5 – 4x 4 + 4x 3 + 10x 2 – 13x – 14. SOLUTION STEP 1 Find the rational zeros of f. Because f is a polynomial function.

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EXAMPLE 2 Find all zeros of f (x) = x 5 – 4x 4 + 4x x 2 – 13x – 14. SOLUTION STEP 1 Find the rational zeros of f. Because f is a polynomial function of degree 5, it has 5 zeros. The possible rational zeros are + 1, + 2, + 7, and Using synthetic division, you can determine that –1 is a zero repeated twice and 2 is also a zero. STEP 2Write f (x) in factored form. Dividing f (x) by its known factors x + 1, x + 1, and x – 2 gives a quotient of x 2 – 4x + 7. Therefore: f (x) = (x + 1) 2 (x – 2)(x 2 – 4x + 7) Find the zeros of a polynomial function

EXAMPLE 2 STEP 3 Find the complex zeros of f. Use the quadratic formula to factor the trinomial into linear factors. f(x) = (x + 1) 2 (x – 2) x – (2 + i 3 ) x – (2 – i 3 ) The zeros of f are –1, –1, 2, 2 + i 3, and 2 – i 3. ANSWER Find the zeros of a polynomial function

GUIDED PRACTICE for Example 2 Find all zeros of the polynomial function. 3. f (x) = x 3 + 7x x + 9 –1 and –3 4. f (x) = x 5 – 2x 4 + 8x 2 – 13x + 6 1, 1, –2, 1 + i 2, and 1 – i 2 ANSWER