Sure, let's factor the polynomial [tex]\( p(x) = x^3 - 2x^2 - 4x^2 + 8x \)[/tex] step by step.
1. Combine like terms:
First, let's simplify the polynomial by combining the like terms:
[tex]\[
p(x) = x^3 - 2x^2 - 4x^2 + 8x
\][/tex]
Notice that we have two terms involving [tex]\( x^2 \)[/tex]. Combine them:
[tex]\[
p(x) = x^3 - 6x^2 + 8x
\][/tex]
2. Factor out the greatest common factor:
Next, we look for the greatest common factor (GCF) in the polynomial. Each term has a common factor of [tex]\( x \)[/tex]:
[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 +8 and add to -6. Those numbers are -2 and -4:
[tex]\[
x^2 - 6x + 8 = (x - 4)(x - 2)
\][/tex]
4. Combine all the factors:
Now we combine all the factors:
[tex]\[
p(x) = x(x - 4)(x - 2)
\][/tex]
So, the factors of the polynomial [tex]\( p(x) = x^3 - 2x^2 - 4x^2 + 8x \)[/tex] are:
[tex]\[
x(x - 4)(x - 2)
\][/tex]
Therefore, the factored form of the polynomial is:
[tex]\[
x(x - 4)(x - 2)
\][/tex]
