Answer :
To factor the polynomial [tex]\(9x^4 + 45x^3 - 9x^2\)[/tex] completely, we'll follow these steps:
1. Identify the Greatest Common Factor (GCF):
First, look for the greatest common factor of all the terms in the polynomial. The terms are [tex]\(9x^4\)[/tex], [tex]\(45x^3\)[/tex], and [tex]\(-9x^2\)[/tex].
The coefficients are 9, 45, and -9, and the common factor among them is 9.
Also, in terms of the variable [tex]\(x\)[/tex], the smallest power of [tex]\(x\)[/tex] is [tex]\(x^2\)[/tex].
Therefore, the GCF of the entire polynomial is [tex]\(9x^2\)[/tex].
2. Factor out the GCF:
Divide each term by the GCF [tex]\(9x^2\)[/tex].
[tex]\[
9x^4 \div 9x^2 = x^2
\][/tex]
[tex]\[
45x^3 \div 9x^2 = 5x
\][/tex]
[tex]\[
-9x^2 \div 9x^2 = -1
\][/tex]
After factoring out [tex]\(9x^2\)[/tex], the polynomial becomes:
[tex]\[
9x^2(x^2 + 5x - 1)
\][/tex]
3. Factor the Remaining Polynomial:
Now, consider the quadratic [tex]\(x^2 + 5x - 1\)[/tex]. We should check if this quadratic can be factored further.
In this case, [tex]\(x^2 + 5x - 1\)[/tex] does not factor nicely over the integers, so it remains as is.
Thus, the final factored form of the polynomial is:
[tex]\[
9x^2(x^2 + 5x - 1)
\][/tex]
This is the complete factorization of the given polynomial.
1. Identify the Greatest Common Factor (GCF):
First, look for the greatest common factor of all the terms in the polynomial. The terms are [tex]\(9x^4\)[/tex], [tex]\(45x^3\)[/tex], and [tex]\(-9x^2\)[/tex].
The coefficients are 9, 45, and -9, and the common factor among them is 9.
Also, in terms of the variable [tex]\(x\)[/tex], the smallest power of [tex]\(x\)[/tex] is [tex]\(x^2\)[/tex].
Therefore, the GCF of the entire polynomial is [tex]\(9x^2\)[/tex].
2. Factor out the GCF:
Divide each term by the GCF [tex]\(9x^2\)[/tex].
[tex]\[
9x^4 \div 9x^2 = x^2
\][/tex]
[tex]\[
45x^3 \div 9x^2 = 5x
\][/tex]
[tex]\[
-9x^2 \div 9x^2 = -1
\][/tex]
After factoring out [tex]\(9x^2\)[/tex], the polynomial becomes:
[tex]\[
9x^2(x^2 + 5x - 1)
\][/tex]
3. Factor the Remaining Polynomial:
Now, consider the quadratic [tex]\(x^2 + 5x - 1\)[/tex]. We should check if this quadratic can be factored further.
In this case, [tex]\(x^2 + 5x - 1\)[/tex] does not factor nicely over the integers, so it remains as is.
Thus, the final factored form of the polynomial is:
[tex]\[
9x^2(x^2 + 5x - 1)
\][/tex]
This is the complete factorization of the given polynomial.
