Answer :
To find the complex zeros of the polynomial function [tex]\( f(x) = x^3 - 15x^2 + 79x - 145 \)[/tex], we can follow a systematic process.
1. Identify the Polynomial:
We have the cubic polynomial:
[tex]\[
f(x) = x^3 - 15x^2 + 79x - 145
\][/tex]
2. Finding the Zeros:
The roots of this polynomial are the values of [tex]\( x \)[/tex] for which [tex]\( f(x) = 0 \)[/tex]. These roots can either be real or complex numbers. For this polynomial, the zeros are:
[tex]\[
x = 5, \quad x = 5 - 2i, \quad x = 5 + 2i
\][/tex]
These roots are what make the polynomial equal zero.
3. Writing the Polynomial in Factored Form:
To express the polynomial in its factored form using these zeros, we take each root and construct a factor. If [tex]\( r \)[/tex] is a root, then [tex]\( (x - r) \)[/tex] is a factor of the polynomial. Therefore, the factored form of the polynomial is:
[tex]\[
f(x) = (x - 5)(x - (5 - 2i))(x - (5 + 2i))
\][/tex]
Since complex roots come in conjugate pairs for polynomials with real coefficients, notice how [tex]\( 5 - 2i \)[/tex] and [tex]\( 5 + 2i \)[/tex] are conjugates.
4. Simplification:
The quadratic term formed by the complex roots is:
[tex]\[
(x - (5 - 2i))(x - (5 + 2i)) = (x - 5 + 2i)(x - 5 - 2i)
\][/tex]
Using the difference of squares formula, this simplifies to:
[tex]\[
(x - 5)^2 - (2i)^2 = (x - 5)^2 + 4
\][/tex]
This further expands to:
[tex]\[
x^2 - 10x + 25 + 4 = x^2 - 10x + 29
\][/tex]
Therefore, the entire polynomial in factored form is:
[tex]\[
f(x) = (x - 5)(x^2 - 10x + 29)
\][/tex]
Thus, the complex zeros of [tex]\( f \)[/tex] are [tex]\( 5 \)[/tex], [tex]\( 5 - 2i \)[/tex], and [tex]\( 5 + 2i \)[/tex]. The polynomial [tex]\( f(x) \)[/tex] in its factored form is:
[tex]\[
f(x) = (x - 5)(x^2 - 10x + 29)
\][/tex]
1. Identify the Polynomial:
We have the cubic polynomial:
[tex]\[
f(x) = x^3 - 15x^2 + 79x - 145
\][/tex]
2. Finding the Zeros:
The roots of this polynomial are the values of [tex]\( x \)[/tex] for which [tex]\( f(x) = 0 \)[/tex]. These roots can either be real or complex numbers. For this polynomial, the zeros are:
[tex]\[
x = 5, \quad x = 5 - 2i, \quad x = 5 + 2i
\][/tex]
These roots are what make the polynomial equal zero.
3. Writing the Polynomial in Factored Form:
To express the polynomial in its factored form using these zeros, we take each root and construct a factor. If [tex]\( r \)[/tex] is a root, then [tex]\( (x - r) \)[/tex] is a factor of the polynomial. Therefore, the factored form of the polynomial is:
[tex]\[
f(x) = (x - 5)(x - (5 - 2i))(x - (5 + 2i))
\][/tex]
Since complex roots come in conjugate pairs for polynomials with real coefficients, notice how [tex]\( 5 - 2i \)[/tex] and [tex]\( 5 + 2i \)[/tex] are conjugates.
4. Simplification:
The quadratic term formed by the complex roots is:
[tex]\[
(x - (5 - 2i))(x - (5 + 2i)) = (x - 5 + 2i)(x - 5 - 2i)
\][/tex]
Using the difference of squares formula, this simplifies to:
[tex]\[
(x - 5)^2 - (2i)^2 = (x - 5)^2 + 4
\][/tex]
This further expands to:
[tex]\[
x^2 - 10x + 25 + 4 = x^2 - 10x + 29
\][/tex]
Therefore, the entire polynomial in factored form is:
[tex]\[
f(x) = (x - 5)(x^2 - 10x + 29)
\][/tex]
Thus, the complex zeros of [tex]\( f \)[/tex] are [tex]\( 5 \)[/tex], [tex]\( 5 - 2i \)[/tex], and [tex]\( 5 + 2i \)[/tex]. The polynomial [tex]\( f(x) \)[/tex] in its factored form is:
[tex]\[
f(x) = (x - 5)(x^2 - 10x + 29)
\][/tex]
