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Answer :
To factor the polynomial [tex]\(-2x^4 + 20x^3 + 48x^2\)[/tex] completely, follow these steps:
1. Identify the Greatest Common Factor (GCF):
First, look for the greatest common factor in all terms of the polynomial. In this case, the GCF of [tex]\(-2x^4\)[/tex], [tex]\(20x^3\)[/tex], and [tex]\(48x^2\)[/tex] is [tex]\(-2x^2\)[/tex].
2. Factor out the GCF:
When you factor [tex]\(-2x^2\)[/tex] out of each term, you get:
[tex]\[
-2x^2(x^2 - 10x - 24)
\][/tex]
3. Factor the quadratic expression inside the parentheses:
Now, focus on factoring the quadratic [tex]\(x^2 - 10x - 24\)[/tex]. To do this, look for two numbers that multiply to [tex]\(-24\)[/tex] (the constant term) and add to [tex]\(-10\)[/tex] (the coefficient of the [tex]\(x\)[/tex] term).
The numbers [tex]\(-12\)[/tex] and [tex]\(2\)[/tex] satisfy this condition because [tex]\(-12 \times 2 = -24\)[/tex] and [tex]\(-12 + 2 = -10\)[/tex].
4. Write the factored form of the quadratic:
The quadratic [tex]\(x^2 - 10x - 24\)[/tex] can be factored into:
[tex]\[
(x - 12)(x + 2)
\][/tex]
5. Combine the factors:
After factoring the quadratic, combine it with the GCF you factored out in the first step:
[tex]\[
-2x^2(x - 12)(x + 2)
\][/tex]
So, the fully factored form of the polynomial [tex]\(-2x^4 + 20x^3 + 48x^2\)[/tex] is:
[tex]\[
-2x^2(x - 12)(x + 2)
\][/tex]
1. Identify the Greatest Common Factor (GCF):
First, look for the greatest common factor in all terms of the polynomial. In this case, the GCF of [tex]\(-2x^4\)[/tex], [tex]\(20x^3\)[/tex], and [tex]\(48x^2\)[/tex] is [tex]\(-2x^2\)[/tex].
2. Factor out the GCF:
When you factor [tex]\(-2x^2\)[/tex] out of each term, you get:
[tex]\[
-2x^2(x^2 - 10x - 24)
\][/tex]
3. Factor the quadratic expression inside the parentheses:
Now, focus on factoring the quadratic [tex]\(x^2 - 10x - 24\)[/tex]. To do this, look for two numbers that multiply to [tex]\(-24\)[/tex] (the constant term) and add to [tex]\(-10\)[/tex] (the coefficient of the [tex]\(x\)[/tex] term).
The numbers [tex]\(-12\)[/tex] and [tex]\(2\)[/tex] satisfy this condition because [tex]\(-12 \times 2 = -24\)[/tex] and [tex]\(-12 + 2 = -10\)[/tex].
4. Write the factored form of the quadratic:
The quadratic [tex]\(x^2 - 10x - 24\)[/tex] can be factored into:
[tex]\[
(x - 12)(x + 2)
\][/tex]
5. Combine the factors:
After factoring the quadratic, combine it with the GCF you factored out in the first step:
[tex]\[
-2x^2(x - 12)(x + 2)
\][/tex]
So, the fully factored form of the polynomial [tex]\(-2x^4 + 20x^3 + 48x^2\)[/tex] is:
[tex]\[
-2x^2(x - 12)(x + 2)
\][/tex]
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