Answer :
To factor the expression [tex]\(-9x^4 + 45x^3 - 9x^2\)[/tex], we can follow these steps:
1. Identify the greatest common factor (GCF):
Look for the GCF of all the terms in the expression [tex]\(-9x^4, 45x^3, -9x^2\)[/tex].
- The GCF of the numerical coefficients [tex]\(-9, 45, -9\)[/tex] is 9.
- The smallest power of [tex]\(x\)[/tex] in the terms is [tex]\(x^2\)[/tex].
So, the GCF is [tex]\(-9x^2\)[/tex].
2. Factor out the GCF:
Extract [tex]\(-9x^2\)[/tex] from each term of the expression:
[tex]\[
-9x^4 + 45x^3 - 9x^2 = -9x^2(x^2 - 5x + 1).
\][/tex]
3. Check the factored expression:
- Ensure that the expression inside the parentheses, [tex]\(x^2 - 5x + 1\)[/tex], cannot be factored further using rational numbers.
- In this case, [tex]\(x^2 - 5x + 1\)[/tex] is not easily factorable further using integers.
Therefore, the given expression [tex]\(-9x^4 + 45x^3 - 9x^2\)[/tex] factors into [tex]\(-9x^2(x^2 - 5x + 1)\)[/tex].
1. Identify the greatest common factor (GCF):
Look for the GCF of all the terms in the expression [tex]\(-9x^4, 45x^3, -9x^2\)[/tex].
- The GCF of the numerical coefficients [tex]\(-9, 45, -9\)[/tex] is 9.
- The smallest power of [tex]\(x\)[/tex] in the terms is [tex]\(x^2\)[/tex].
So, the GCF is [tex]\(-9x^2\)[/tex].
2. Factor out the GCF:
Extract [tex]\(-9x^2\)[/tex] from each term of the expression:
[tex]\[
-9x^4 + 45x^3 - 9x^2 = -9x^2(x^2 - 5x + 1).
\][/tex]
3. Check the factored expression:
- Ensure that the expression inside the parentheses, [tex]\(x^2 - 5x + 1\)[/tex], cannot be factored further using rational numbers.
- In this case, [tex]\(x^2 - 5x + 1\)[/tex] is not easily factorable further using integers.
Therefore, the given expression [tex]\(-9x^4 + 45x^3 - 9x^2\)[/tex] factors into [tex]\(-9x^2(x^2 - 5x + 1)\)[/tex].
