College

Factor the trinomial completely:

[tex]x^5 - 8x^4 - 48x^3[/tex]

Select the correct choice below and, if necessary, fill in the answer box to complete your choice.

A. [tex]x^5 - 8x^4 - 48x^3 =[/tex] [tex]\square[/tex] (Factor completely.)

B. The polynomial is prime.

Answer :

To factor the trinomial [tex]\( x^5 - 8x^4 - 48x^3 \)[/tex] completely, follow these step-by-step instructions:

1. Identify the Greatest Common Factor (GCF):

First, look at all the terms in the trinomial. Notice that each term contains at least [tex]\( x^3 \)[/tex]. So, the GCF is [tex]\( x^3 \)[/tex].

2. Factor out the GCF:

Factor out [tex]\( x^3 \)[/tex] from the entire expression:

[tex]\[
x^5 - 8x^4 - 48x^3 = x^3(x^2 - 8x - 48)
\][/tex]

3. Factor the Quadratic Expression:

Now, focus on the quadratic expression inside the parentheses: [tex]\( x^2 - 8x - 48 \)[/tex]. We need to factor this quadratic expression further.

To factor [tex]\( x^2 - 8x - 48 \)[/tex], look for two numbers that multiply to -48 (the constant term) and add to -8 (the coefficient of the middle term). These two numbers are -12 and 4.

4. Write the Quadratic in Factored Form:

This means we can write the quadratic expression as the product of two binomials:

[tex]\[
x^2 - 8x - 48 = (x - 12)(x + 4)
\][/tex]

5. Combine Everything Together:

Combine the factored quadratic with the GCF we factored out initially:

[tex]\[
x^5 - 8x^4 - 48x^3 = x^3(x - 12)(x + 4)
\][/tex]

This is the trinomial completely factored. The correct answer is:

[tex]\[
x^3(x - 12)(x + 4)
\][/tex]

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