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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]
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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