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Answer :
Sure! Let's factor the polynomial [tex]\(-2x^4 + 20x^3 + 48x^2\)[/tex] completely, step by step.
1. Identify the greatest common factor (GCF):
Start by looking for the greatest common factor of all the terms. In this polynomial, each term has a common factor of [tex]\(-2x^2\)[/tex].
2. Factor out the GCF:
Extract [tex]\(-2x^2\)[/tex] from all the terms:
[tex]\[
-2x^4 + 20x^3 + 48x^2 = -2x^2(x^2 - 10x - 24)
\][/tex]
3. Factor the quadratic expression:
Now, focus on the quadratic expression inside the parentheses: [tex]\(x^2 - 10x - 24\)[/tex].
4. Find two numbers that multiply to [tex]\(-24\)[/tex] (the constant term) and add to [tex]\(-10\)[/tex] (the linear coefficient):
The numbers that work here are [tex]\(-12\)[/tex] and [tex]\(2\)[/tex], because:
[tex]\[
-12 \times 2 = -24 \quad \text{and} \quad -12 + 2 = -10
\][/tex]
5. Write the quadratic expression as a product of two binomials:
Using the numbers found in the previous step, we can factor the quadratic as:
[tex]\[
x^2 - 10x - 24 = (x - 12)(x + 2)
\][/tex]
6. Combine the factors:
Now, substitute the factored quadratic expression back into the original expression:
[tex]\[
-2x^2(x^2 - 10x - 24) = -2x^2(x - 12)(x + 2)
\][/tex]
So, the completely 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):
Start by looking for the greatest common factor of all the terms. In this polynomial, each term has a common factor of [tex]\(-2x^2\)[/tex].
2. Factor out the GCF:
Extract [tex]\(-2x^2\)[/tex] from all the terms:
[tex]\[
-2x^4 + 20x^3 + 48x^2 = -2x^2(x^2 - 10x - 24)
\][/tex]
3. Factor the quadratic expression:
Now, focus on the quadratic expression inside the parentheses: [tex]\(x^2 - 10x - 24\)[/tex].
4. Find two numbers that multiply to [tex]\(-24\)[/tex] (the constant term) and add to [tex]\(-10\)[/tex] (the linear coefficient):
The numbers that work here are [tex]\(-12\)[/tex] and [tex]\(2\)[/tex], because:
[tex]\[
-12 \times 2 = -24 \quad \text{and} \quad -12 + 2 = -10
\][/tex]
5. Write the quadratic expression as a product of two binomials:
Using the numbers found in the previous step, we can factor the quadratic as:
[tex]\[
x^2 - 10x - 24 = (x - 12)(x + 2)
\][/tex]
6. Combine the factors:
Now, substitute the factored quadratic expression back into the original expression:
[tex]\[
-2x^2(x^2 - 10x - 24) = -2x^2(x - 12)(x + 2)
\][/tex]
So, the completely 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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