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Find the complex zeros of the following polynomial function. Write [tex]f[/tex] in factored form.

[tex]f(x) = x^4 + 170x^2 + 169[/tex]

The complex zeros of [tex]f[/tex] are [tex]\square[/tex].

(Simplify your answer. Type an exact answer, using radicals and [tex]i[/tex] as needed. Use integers.)

Answer :

To find the complex zeros of the polynomial [tex]\( f(x) = x^4 + 170x^2 + 169 \)[/tex] and write it in factored form, we can utilize substitution and solve it step by step.

1. Substitute [tex]\( y = x^2 \)[/tex]:
- If we let [tex]\( y = x^2 \)[/tex], then the polynomial becomes:
[tex]\[
f(y) = y^2 + 170y + 169
\][/tex]
- This is now a quadratic in terms of [tex]\( y \)[/tex].

2. Solve the quadratic equation [tex]\( y^2 + 170y + 169 = 0 \)[/tex]:
- Use the quadratic formula to solve for [tex]\( y \)[/tex]:
[tex]\[
y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
where [tex]\( a = 1 \)[/tex], [tex]\( b = 170 \)[/tex], and [tex]\( c = 169 \)[/tex].
- Calculate the discriminant:
[tex]\[
b^2 - 4ac = 170^2 - 4 \times 1 \times 169 = 28900 - 676 = 28224
\][/tex]
- Find the square root of the discriminant:
[tex]\[
\sqrt{28224} = 168
\][/tex]
- Plug these values into the quadratic formula:
[tex]\[
y = \frac{-170 \pm 168}{2}
\][/tex]
- This gives us two solutions:
[tex]\[
y_1 = \frac{-170 + 168}{2} = -1 \quad \text{and} \quad y_2 = \frac{-170 - 168}{2} = -169
\][/tex]

3. Find [tex]\( x \)[/tex] from [tex]\( y = x^2 \)[/tex]:
- For [tex]\( y = -1 \)[/tex]:
[tex]\[
x^2 = -1 \implies x = \pm i
\][/tex]
- For [tex]\( y = -169 \)[/tex]:
[tex]\[
x^2 = -169 \implies x = \pm 13i
\][/tex]

4. List the complex zeros:
- The complex zeros are [tex]\( i, -i, 13i, \)[/tex] and [tex]\( -13i \)[/tex].

5. Write [tex]\( f(x) \)[/tex] in factored form:
- Given these roots, the factored form of [tex]\( f(x) \)[/tex] is:
[tex]\[
f(x) = (x - i)(x + i)(x - 13i)(x + 13i)
\][/tex]

So, the complex zeros of the polynomial [tex]\( f(x) = x^4 + 170x^2 + 169 \)[/tex] are [tex]\( i, -i, 13i, \)[/tex] and [tex]\( -13i \)[/tex].

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