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Answer :
To find the complex zeros of the polynomial function [tex]\( f(x) = x^4 + 170x^2 + 169 \)[/tex] and write [tex]\( f \)[/tex] in factored form, follow these steps:
1. Understanding the Polynomial: We have a polynomial [tex]\( f(x) = x^4 + 170x^2 + 169 \)[/tex].
2. Think of a Substitution: Notice that the polynomial has terms with even powers of [tex]\( x \)[/tex]. We can use a substitution to simplify it. Let [tex]\( y = x^2 \)[/tex]. Then the polynomial becomes:
[tex]\[
f(x) = y^2 + 170y + 169
\][/tex]
3. Solve the Quadratic Equation: Now, solve the quadratic equation [tex]\( y^2 + 170y + 169 = 0 \)[/tex] using the quadratic formula:
[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].
4. Calculate the Discriminant: First, compute the discriminant:
[tex]\[
b^2 - 4ac = 170^2 - 4 \cdot 1 \cdot 169 = 28900 - 676 = 28224
\][/tex]
[tex]\[
\sqrt{28224} = 168
\][/tex]
5. Find the Roots of the Quadratic: Substitute back into the quadratic formula:
[tex]\[
y = \frac{-170 \pm 168}{2}
\][/tex]
This gives us the solutions:
[tex]\[
y_1 = \frac{-170 + 168}{2} = -1
\][/tex]
[tex]\[
y_2 = \frac{-170 - 168}{2} = -169
\][/tex]
6. Find [tex]\( x \)[/tex] for Each Solution: Since [tex]\( y = x^2 \)[/tex], solve for [tex]\( x \)[/tex]:
- For [tex]\( y = -1 \)[/tex], [tex]\( x^2 = -1 \)[/tex], giving solutions:
[tex]\[
x = \pm i
\][/tex]
- For [tex]\( y = -169 \)[/tex], [tex]\( x^2 = -169 \)[/tex], giving solutions:
[tex]\[
x = \pm 13i
\][/tex]
These give us the complex zeros: [tex]\( x = i, -i, 13i, -13i \)[/tex].
7. Write the Polynomial in Factored Form: Using the zeros found, write the polynomial in factored form:
[tex]\[
f(x) = (x - i)(x + i)(x - 13i)(x + 13i)
\][/tex]
Simplifying this gives:
[tex]\[
f(x) = (x^2 + 1)(x^2 + 169)
\][/tex]
Thus, the complex zeros of [tex]\( f \)[/tex] are [tex]\( i, -i, 13i, \)[/tex] and [tex]\( -13i \)[/tex], and the factored form of the polynomial is [tex]\( (x^2 + 1)(x^2 + 169) \)[/tex].
1. Understanding the Polynomial: We have a polynomial [tex]\( f(x) = x^4 + 170x^2 + 169 \)[/tex].
2. Think of a Substitution: Notice that the polynomial has terms with even powers of [tex]\( x \)[/tex]. We can use a substitution to simplify it. Let [tex]\( y = x^2 \)[/tex]. Then the polynomial becomes:
[tex]\[
f(x) = y^2 + 170y + 169
\][/tex]
3. Solve the Quadratic Equation: Now, solve the quadratic equation [tex]\( y^2 + 170y + 169 = 0 \)[/tex] using the quadratic formula:
[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].
4. Calculate the Discriminant: First, compute the discriminant:
[tex]\[
b^2 - 4ac = 170^2 - 4 \cdot 1 \cdot 169 = 28900 - 676 = 28224
\][/tex]
[tex]\[
\sqrt{28224} = 168
\][/tex]
5. Find the Roots of the Quadratic: Substitute back into the quadratic formula:
[tex]\[
y = \frac{-170 \pm 168}{2}
\][/tex]
This gives us the solutions:
[tex]\[
y_1 = \frac{-170 + 168}{2} = -1
\][/tex]
[tex]\[
y_2 = \frac{-170 - 168}{2} = -169
\][/tex]
6. Find [tex]\( x \)[/tex] for Each Solution: Since [tex]\( y = x^2 \)[/tex], solve for [tex]\( x \)[/tex]:
- For [tex]\( y = -1 \)[/tex], [tex]\( x^2 = -1 \)[/tex], giving solutions:
[tex]\[
x = \pm i
\][/tex]
- For [tex]\( y = -169 \)[/tex], [tex]\( x^2 = -169 \)[/tex], giving solutions:
[tex]\[
x = \pm 13i
\][/tex]
These give us the complex zeros: [tex]\( x = i, -i, 13i, -13i \)[/tex].
7. Write the Polynomial in Factored Form: Using the zeros found, write the polynomial in factored form:
[tex]\[
f(x) = (x - i)(x + i)(x - 13i)(x + 13i)
\][/tex]
Simplifying this gives:
[tex]\[
f(x) = (x^2 + 1)(x^2 + 169)
\][/tex]
Thus, the complex zeros of [tex]\( f \)[/tex] are [tex]\( i, -i, 13i, \)[/tex] and [tex]\( -13i \)[/tex], and the factored form of the polynomial is [tex]\( (x^2 + 1)(x^2 + 169) \)[/tex].
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