GUIDED PRACTICE for Example 1

1. How many solutions does the equation x4 + 5x2 – 36 = 0 have?

ANSWER 4

GUIDED PRACTICE for Example 1

2. How many zeros does the function f (x) = x3 + 7x2 + 8x – 16 have?

ANSWER 3

GUIDED PRACTICE for Example 2

Find all zeros of the polynomial function.

3. f (x) = x3 + 7x2 + 15x + 9

STEP 1 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 2 Write f (x) in factored form

Formula are (x +1)2 (x +3)

f(x) = (x +1) (x +3)2

The zeros of f are – 1 and – 3

SOLUTION

GUIDED PRACTICE for Example 2

4. f (x) = x5 – 2x4 + 8x2 – 13x + 6

STEP 1 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

+– +– +–+– 6.

STEP 2 Write f (x) in factored form dividing f(x)by its known factor (x – 1),(x – 1)and (x+2) given a qualities x2 – 2x +3 therefore

f (x) = (x – 1)2 (x+2) (x2 – 2x + 3)

SOLUTION

GUIDED PRACTICE for Example 2

Find the complex zero of f. use the quadratic formula to factor the trinomial into linear factor

STEP 3

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

= (x + 1) (x2 – 4x – 2x + 8)

f (x) = (x + 1) (x – 2) ( x – 4)

GUIDED PRACTICE

Write a polynomial function f of least degree that has rational coefficients, a leading coefficient of 1, and the given zeros.

5. – 1, 2, 4

for Example 3

Write f (x) in factored form.

= (x + 1) (x2 – 6x + 8)

Multiply.= x3 – 6x2 + 8x + x2 – 6x + 8= x3 – 5x2 + 2x + 8

Multiply.

Combine like terms.

Combine like terms.

Use the three zeros and the factor theorem to write f(x) as a product of three factors.SOLUTION

GUIDED PRACTICE for Example 3

6. 4, 1 + √ 5

f (x) = (x – 4) [ x – (1 + √ 5 ) ] [ x – (1 – √ 5 ) ]Write f (x) in factored form.Regroup terms.= (x – 4) [ (x – 1) – √ 5 ] [ (x – 1) +√ 5 ]

Multiply.

= (x – 4)[(x2 – 2x + 1) – 5] Expand binomial.

= (x – 4)[(x – 1)2 – ( 5)2]

Because the coefficients are rational and 1 + 5 is a zero, 1 – 5 must also be a zero by the irrational conjugates theorem. Use the three zeros and the factor theorem to write f (x) as a product of three factors

SOLUTION

GUIDED PRACTICE for Example 3

= (x – 4)(x2 – 2x – 4) Simplify.

= x3 – 2x2 – 4x – 4x2 + 8x + 16 Multiply.

= x3 – 6x2 + 4x +16 Combine like terms.

GUIDED PRACTICE

7. 2, 2i, 4 – √ 6

for Example 3

Because the coefficients are rational and 2i is a zero, –2i must also be a zero by the complex conjugates theorem. 4 + 6 is also a zero by the irrational conjugate theorem. Use the five zeros and the factor theorem to write f(x) as a product of five factors.

f (x) = (x–2) (x +2i)(x-2i)[(x –(4 –√6 )][x –(4+√6) ] Write f (x) in factored form.Regroup terms.= (x – 2) [ (x2 –(2i)2][x2–4)+√6][(x– 4) – √6 ]

Multiply.

= (x – 2)(x2 + 4)(x2 – 8x+16 – 6) Expand binomial.

= (x – 2)[(x2 + 4)[(x– 4)2 – ( 6 )2]

SOLUTION

GUIDED PRACTICE for Example 3

= (x – 2)(x2 + 4)(x2 – 8x + 10) Simplify.

= (x–2) (x4– 8x2 +10x2 +4x2 –3x +40) Multiply.

= (x–2) (x4 – 8x3 +14x2 –32x + 40) Combine like terms.

= x5– 8x4 +14x3 –32x2 +40x – 2x4

+16x3 –28x2 + 64x – 80

= x5–10x4 + 30x3 – 60x2 +10x – 80 Combine like terms.

Multiply.

GUIDED PRACTICE

8. 3, 3 – i

for Example 3

Because the coefficients are rational and 3 –i is a zero, 3 + i must also be a zero by the complex conjugates theorem. Use the three zeros and the factor theorem to write f(x) as a product of three factors

= f(x) =(x – 3)[x – (3 – i)][x –(3 + i)]

= (x–3)[(x– 3)+i ][(x – 3) – i] Regroup terms.

= (x–3)[(x – 3)2 –i2)]

= (x– 3)[(x – 3)+ i][(x –3) –i]

Multiply.

Write f (x) in factored form.

SOLUTION

GUIDED PRACTICE for Example 3

= (x – 3)[(x – 3)2 – i2]=(x –3)(x2 – 6x + 9)

Simplify.= (x–3)(x2 – 6x + 9)

= x3–6x2 + 9x – 3x2 +18x – 27

Combine like terms.= x3 – 9x2 + 27x –27

Multiply.