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Math  /  Algebra

QuestionFind the complex zeros of the following polynomial function. Write ff in factored form. f(x)=2x4−13x3−16x2+243x−116f(x)=2 x^{4}-13 x^{3}-16 x^{2}+243 x-116
The complex zeros of f are 12,−4,5+2i,5−2i\frac{1}{2},-4,5+2 i, 5-2 i. (Simplify your answer. Type an exact answer, using radicals and ii as needed. Use integers or fractions for any nu separate answers as needed.)
Use the complex zeros to factor f. f(x)=f(x)= □\square

Studdy Solution

STEP 1

1. We are given the polynomial function f(x)=2x4−13x3−16x2+243x−116 f(x) = 2x^4 - 13x^3 - 16x^2 + 243x - 116 .
2. The complex zeros of f f are 12,−4,5+2i,5−2i \frac{1}{2}, -4, 5+2i, 5-2i .
3. We need to write f f in its factored form using these zeros.

STEP 2

1. Use the given zeros to write the factors of the polynomial.
2. Write the polynomial in factored form.
3. Verify the factored form by expanding and comparing to the original polynomial.

STEP 3

Each zero c c of a polynomial corresponds to a factor of the form (x−c) (x - c) .
Given zeros: 12,−4,5+2i,5−2i \frac{1}{2}, -4, 5+2i, 5-2i .
The factors corresponding to these zeros are: - (x−12) \left(x - \frac{1}{2}\right) - (x+4) (x + 4) - (x−(5+2i)) (x - (5 + 2i)) - (x−(5−2i)) (x - (5 - 2i))

STEP 4

Combine the factors to write the polynomial in factored form:
f(x)=2(x−12)(x+4)(x−(5+2i))(x−(5−2i)) f(x) = 2 \left(x - \frac{1}{2}\right)(x + 4)(x - (5 + 2i))(x - (5 - 2i))
Since 5+2i 5+2i and 5−2i 5-2i are complex conjugates, their product is a quadratic with real coefficients:
(x−(5+2i))(x−(5−2i))=(x−5)2−(2i)2 (x - (5 + 2i))(x - (5 - 2i)) = (x - 5)^2 - (2i)^2 =(x−5)2−(−4) = (x - 5)^2 - (-4) =(x−5)2+4 = (x - 5)^2 + 4

STEP 5

Now, write the complete factored form of f(x) f(x) :
f(x)=2(x−12)(x+4)((x−5)2+4) f(x) = 2 \left(x - \frac{1}{2}\right)(x + 4)((x - 5)^2 + 4)
Simplify the factor (x−12) \left(x - \frac{1}{2}\right) :
2(x−12)=(2x−1) 2 \left(x - \frac{1}{2}\right) = (2x - 1)
Thus, the factored form is:
f(x)=(2x−1)(x+4)((x−5)2+4) f(x) = (2x - 1)(x + 4)((x - 5)^2 + 4)

STEP 6

Verify by expanding the factors and comparing to the original polynomial f(x)=2x4−13x3−16x2+243x−116 f(x) = 2x^4 - 13x^3 - 16x^2 + 243x - 116 .
Expand (x−5)2+4 (x - 5)^2 + 4 :
(x−5)2=x2−10x+25 (x - 5)^2 = x^2 - 10x + 25 (x−5)2+4=x2−10x+29 (x - 5)^2 + 4 = x^2 - 10x + 29
Now expand (2x−1)(x+4)(x2−10x+29) (2x - 1)(x + 4)(x^2 - 10x + 29) and verify it matches the original polynomial.
The factored form of f(x) f(x) is:
(2x−1)(x+4)((x−5)2+4) \boxed{(2x - 1)(x + 4)((x - 5)^2 + 4)}

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