Answer :
To factor the polynomial [tex]\(-2x^4 + 20x^3 + 48x^2\)[/tex] completely, we'll go through the steps, starting with finding the greatest common factor (GCF).
### Step 1: Find the Greatest Common Factor (GCF)
The given polynomial is:
[tex]\[
-2x^4 + 20x^3 + 48x^2
\][/tex]
First, identify the GCF of the coefficients and the smallest power of [tex]\(x\)[/tex] present in all terms.
- The coefficients are [tex]\(-2\)[/tex], [tex]\(20\)[/tex], and [tex]\(48\)[/tex]. The GCF of these numbers is 2.
- Each term includes [tex]\(x\)[/tex], and the smallest power is [tex]\(x^2\)[/tex].
Thus, the GCF of the entire polynomial is [tex]\(2x^2\)[/tex].
### Step 2: Factor Out the GCF
Factor out the GCF from each term:
[tex]\[
-2x^4 + 20x^3 + 48x^2 = 2x^2(-x^2 + 10x + 24)
\][/tex]
### Step 3: Factor the Remaining Polynomial
Now we focus on factoring the quadratic polynomial inside the parentheses: [tex]\(-x^2 + 10x + 24\)[/tex].
1. Rearrange for Easier Factorization:
To make it easier to factor, we can take out a negative sign from [tex]\(-x^2 + 10x + 24\)[/tex]:
[tex]\[
-1(x^2 - 10x - 24)
\][/tex]
2. Factor the Quadratic Expression:
We now factor [tex]\(x^2 - 10x - 24\)[/tex]. We need two numbers that multiply to [tex]\(-24\)[/tex] and add up to [tex]\(-10\)[/tex].
The numbers are [tex]\(12\)[/tex] and [tex]\(-2\)[/tex]:
[tex]\[
x^2 - 10x - 24 = (x - 12)(x + 2)
\][/tex]
3. Incorporate the Negative Sign:
Bring back the negative sign:
[tex]\[
-1 \cdot (x - 12)(x + 2) = -(x - 12)(x + 2)
\][/tex]
### Step 4: Write the Completely Factored Form
Now, substitute back into the expression with the GCF:
[tex]\[
-2x^2 \cdot (x - 12)(x + 2)
\][/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]
This is your final answer after factoring completely!
### Step 1: Find the Greatest Common Factor (GCF)
The given polynomial is:
[tex]\[
-2x^4 + 20x^3 + 48x^2
\][/tex]
First, identify the GCF of the coefficients and the smallest power of [tex]\(x\)[/tex] present in all terms.
- The coefficients are [tex]\(-2\)[/tex], [tex]\(20\)[/tex], and [tex]\(48\)[/tex]. The GCF of these numbers is 2.
- Each term includes [tex]\(x\)[/tex], and the smallest power is [tex]\(x^2\)[/tex].
Thus, the GCF of the entire polynomial is [tex]\(2x^2\)[/tex].
### Step 2: Factor Out the GCF
Factor out the GCF from each term:
[tex]\[
-2x^4 + 20x^3 + 48x^2 = 2x^2(-x^2 + 10x + 24)
\][/tex]
### Step 3: Factor the Remaining Polynomial
Now we focus on factoring the quadratic polynomial inside the parentheses: [tex]\(-x^2 + 10x + 24\)[/tex].
1. Rearrange for Easier Factorization:
To make it easier to factor, we can take out a negative sign from [tex]\(-x^2 + 10x + 24\)[/tex]:
[tex]\[
-1(x^2 - 10x - 24)
\][/tex]
2. Factor the Quadratic Expression:
We now factor [tex]\(x^2 - 10x - 24\)[/tex]. We need two numbers that multiply to [tex]\(-24\)[/tex] and add up to [tex]\(-10\)[/tex].
The numbers are [tex]\(12\)[/tex] and [tex]\(-2\)[/tex]:
[tex]\[
x^2 - 10x - 24 = (x - 12)(x + 2)
\][/tex]
3. Incorporate the Negative Sign:
Bring back the negative sign:
[tex]\[
-1 \cdot (x - 12)(x + 2) = -(x - 12)(x + 2)
\][/tex]
### Step 4: Write the Completely Factored Form
Now, substitute back into the expression with the GCF:
[tex]\[
-2x^2 \cdot (x - 12)(x + 2)
\][/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]
This is your final answer after factoring completely!
