To factor the polynomial function [tex]\( p(x) = x^3 - 2x^2 - 4x^2 + 8x \)[/tex], we will follow these steps:
1. Combine Like Terms: First, let's combine the like terms in the polynomial.
[tex]\[
p(x) = x^3 - 2x^2 - 4x^2 + 8x
\][/tex]
Notice that we can combine the [tex]\( x^2 \)[/tex] terms:
[tex]\[
p(x) = x^3 - (2x^2 + 4x^2) + 8x
\][/tex]
Simplifying the expression inside the parentheses:
[tex]\[
p(x) = x^3 - 6x^2 + 8x
\][/tex]
2. Factor Out the Greatest Common Factor (GCF): Next, we look for the greatest common factor that can be factored out of each term in the polynomial. Each term [tex]\( x^3, -6x^2, \)[/tex] and [tex]\( 8x \)[/tex] has at least one [tex]\( x \)[/tex] in common.
Factoring out [tex]\( x \)[/tex], we get:
[tex]\[
p(x) = x(x^2 - 6x + 8)
\][/tex]
3. Factor the Quadratic Expression: Now we need to factor the quadratic expression [tex]\( x^2 - 6x + 8 \)[/tex].
We look for two numbers that multiply to [tex]\( 8 \)[/tex] (the constant term) and add up to [tex]\( -6 \)[/tex] (the coefficient of [tex]\( x \)[/tex]). These numbers are [tex]\( -2 \)[/tex] and [tex]\( -4 \)[/tex]:
[tex]\[
x^2 - 6x + 8 = (x - 2)(x - 4)
\][/tex]
4. Write the Final Factored Form: Combining these factors, we get the completely factored form of the polynomial:
[tex]\[
p(x) = x(x - 2)(x - 4)
\][/tex]
Thus, the polynomial [tex]\( p(x) = x^3 - 2x^2 - 4x^2 + 8x \)[/tex] factors as:
[tex]\[
x(x - 2)(x - 4)
\][/tex]
