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Polynomial Zeros Calculation Method

Learn how to find rational and complex zeros of polynomial functions step-by-step using synthetic division and factored form. Practice problems included.

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Polynomial Zeros Calculation Method

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  1. STEP 1 Find the rational zeros of f. Because fis a polynomial function of degree 5, it has 5 zeros. The possible rational zeros are + 1, + 2, + 7, and +14. Using synthetic division, you can determine that – 1 is a zero repeated twice and 2 is also a zero. STEP 2 Write f (x) in factored form. Dividing f (x) by its known factors x + 1, x + 1, and x – 2 gives a quotient of x2– 4x + 7. Therefore: EXAMPLE 2 Find the zeros of a polynomial function Find all zeros of f (x) = x5 – 4x4 + 4x3 + 10x2 – 13x – 14. SOLUTION f (x) = (x + 1)2(x – 2)(x2 – 4x + 7)

  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 ) ANSWER The zeros of f are – 1, – 1, 2, 2 + i 3 , and2 – i 3. EXAMPLE 2 Find the zeros of a polynomial function

  3. Find the rational zero of f. because f is a polynomial function degree 3, it has 3 zero. The possible rational zeros are 1 , 3, using synthetic division, you can determine that 3 is a zero reputed twice and –3 is also a zero STEP 1 STEP 2 Writef (x) in factored form + + – – for Example 2 GUIDED PRACTICE Find all zeros of the polynomial function. 3. f (x) = x3 + 7x2 + 15x + 9 SOLUTION Formula are (x +1)2 (x +3) f(x) = (x +1) (x +3)2 The zeros of f are – 1 and – 3

  4. Find the rational zero of f. because f is a polynomial function degree 5, it has 5 zero. The possible rational zeros are 1 , 2, 3 and Using synthetic division, you can determine that 1 is a zero reputed twice and –3 is also a zero STEP 1 6. STEP 2 Writef (x) in factored formdividing f(x)by its known factor (x – 1),(x – 1)and (x+2) given a qualitiesx2 – 2x +3 therefore + + + + – – – – for Example 2 GUIDED PRACTICE 4. f (x) = x5– 2x4 + 8x2– 13x + 6 SOLUTION f (x) = (x – 1)2 (x+2) (x2 – 2x + 3)

  5. STEP 3 Find the complex zero of f. use the quadratic formula to factor the trinomial into linear factor f (x) = (x –1)2 (x + 2) [x – (1 + i 2) [ (x – (1 – i 2)] Zeros of f are 1, 1, – 2, 1 + i 2 , and 1 – i 2 for Example 2 GUIDED PRACTICE

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