Answer :
To factor the expression [tex]\(-5x^4 + 70x^3 + 75x^2\)[/tex] completely, let's break it down step-by-step:
1. Find the Greatest Common Factor (GCF):
Look at the coefficients and the powers of [tex]\(x\)[/tex]. The GCF of the coefficients [tex]\(-5, 70, 75\)[/tex] is 5, and the lowest power of [tex]\(x\)[/tex] is [tex]\(x^2\)[/tex]. So, the GCF of the whole expression is [tex]\(5x^2\)[/tex].
2. Factor Out the GCF:
Divide each term in the expression [tex]\(-5x^4 + 70x^3 + 75x^2\)[/tex] by [tex]\(5x^2\)[/tex].
- [tex]\(-5x^4 \div 5x^2 = -x^2\)[/tex]
- [tex]\(70x^3 \div 5x^2 = 14x\)[/tex]
- [tex]\(75x^2 \div 5x^2 = 15\)[/tex]
This step gives us:
[tex]\(-5x^2(x^2 - 14x + 15)\)[/tex]
3. Factor the Quadratic Expression:
Now factor the quadratic [tex]\(x^2 - 14x + 15\)[/tex].
We need to find two numbers that multiply to 15 and add up to -14. These numbers are -15 and +1.
4. Write the Expression in Factored Form:
Substitute these numbers back into the quadratic. The expression becomes:
[tex]\(x^2 - 14x + 15 = (x - 15)(x + 1)\)[/tex]
5. Combine All Parts Together:
Combine everything from the previous steps:
[tex]\(-5x^2(x - 15)(x + 1)\)[/tex]
So, the completely factored form of the expression [tex]\(-5x^4 + 70x^3 + 75x^2\)[/tex] is [tex]\(-5x^2(x - 15)(x + 1)\)[/tex].
1. Find the Greatest Common Factor (GCF):
Look at the coefficients and the powers of [tex]\(x\)[/tex]. The GCF of the coefficients [tex]\(-5, 70, 75\)[/tex] is 5, and the lowest power of [tex]\(x\)[/tex] is [tex]\(x^2\)[/tex]. So, the GCF of the whole expression is [tex]\(5x^2\)[/tex].
2. Factor Out the GCF:
Divide each term in the expression [tex]\(-5x^4 + 70x^3 + 75x^2\)[/tex] by [tex]\(5x^2\)[/tex].
- [tex]\(-5x^4 \div 5x^2 = -x^2\)[/tex]
- [tex]\(70x^3 \div 5x^2 = 14x\)[/tex]
- [tex]\(75x^2 \div 5x^2 = 15\)[/tex]
This step gives us:
[tex]\(-5x^2(x^2 - 14x + 15)\)[/tex]
3. Factor the Quadratic Expression:
Now factor the quadratic [tex]\(x^2 - 14x + 15\)[/tex].
We need to find two numbers that multiply to 15 and add up to -14. These numbers are -15 and +1.
4. Write the Expression in Factored Form:
Substitute these numbers back into the quadratic. The expression becomes:
[tex]\(x^2 - 14x + 15 = (x - 15)(x + 1)\)[/tex]
5. Combine All Parts Together:
Combine everything from the previous steps:
[tex]\(-5x^2(x - 15)(x + 1)\)[/tex]
So, the completely factored form of the expression [tex]\(-5x^4 + 70x^3 + 75x^2\)[/tex] is [tex]\(-5x^2(x - 15)(x + 1)\)[/tex].
