Answer :
To factor the polynomial [tex]\(-9x^4 + 45x^3 + 18x^2\)[/tex] using the negative of the greatest common factor, we'll follow these steps:
1. Identify the Greatest Common Factor (GCF):
First, look at the coefficients: [tex]\(-9\)[/tex], [tex]\(45\)[/tex], and [tex]\(18\)[/tex]. We want to find the greatest common factor of these numbers:
- The factors of [tex]\(-9\)[/tex] are [tex]\(1, 3, 9, -1, -3, -9\)[/tex].
- The factors of [tex]\(45\)[/tex] are [tex]\(1, 3, 5, 9, 15, 45\)[/tex].
- The factors of [tex]\(18\)[/tex] are [tex]\(1, 2, 3, 6, 9, 18\)[/tex].
The greatest common factor (ignoring sign) is [tex]\(9\)[/tex]. Since we want the negative of the greatest common factor, we use [tex]\(-9\)[/tex].
2. Factor out [tex]\(-9x^2\:
We also notice that each term in the polynomial includes at least \(x^2\)[/tex]. So the greatest common factor of the polynomial includes both the numerical factor [tex]\(-9\)[/tex] and [tex]\(x^2\)[/tex].
Now we will factor [tex]\(-9x^2\)[/tex] out of each term:
- From [tex]\(-9x^4\)[/tex], factoring out [tex]\(-9x^2\)[/tex] leaves us with [tex]\(x^2\)[/tex].
- From [tex]\(45x^3\)[/tex], factoring out [tex]\(-9x^2\)[/tex] leaves us with [tex]\(-5x\)[/tex].
- From [tex]\(18x^2\)[/tex], factoring out [tex]\(-9x^2\)[/tex] leaves us with [tex]\(-2\)[/tex].
After factoring out [tex]\(-9x^2\)[/tex], the polynomial becomes:
[tex]\[
-9x^2(x^2 - 5x - 2)
\][/tex]
3. Check the factored expression:
We can verify by distributing [tex]\(-9x^2\)[/tex] back through the polynomial [tex]\((x^2 - 5x - 2)\)[/tex] to ensure it simplifies back to the original polynomial:
[tex]\[
-9x^2 \cdot x^2 = -9x^4, \quad
-9x^2 \cdot (-5x) = 45x^3, \quad
-9x^2 \cdot (-2) = 18x^2
\][/tex]
The expanded form matches the original polynomial, confirming our factorization is correct.
So, the factorized form of [tex]\(-9x^4 + 45x^3 + 18x^2\)[/tex] using the negative of the greatest common factor is:
[tex]\[
-9x^2(x^2 - 5x - 2)
\][/tex]
1. Identify the Greatest Common Factor (GCF):
First, look at the coefficients: [tex]\(-9\)[/tex], [tex]\(45\)[/tex], and [tex]\(18\)[/tex]. We want to find the greatest common factor of these numbers:
- The factors of [tex]\(-9\)[/tex] are [tex]\(1, 3, 9, -1, -3, -9\)[/tex].
- The factors of [tex]\(45\)[/tex] are [tex]\(1, 3, 5, 9, 15, 45\)[/tex].
- The factors of [tex]\(18\)[/tex] are [tex]\(1, 2, 3, 6, 9, 18\)[/tex].
The greatest common factor (ignoring sign) is [tex]\(9\)[/tex]. Since we want the negative of the greatest common factor, we use [tex]\(-9\)[/tex].
2. Factor out [tex]\(-9x^2\:
We also notice that each term in the polynomial includes at least \(x^2\)[/tex]. So the greatest common factor of the polynomial includes both the numerical factor [tex]\(-9\)[/tex] and [tex]\(x^2\)[/tex].
Now we will factor [tex]\(-9x^2\)[/tex] out of each term:
- From [tex]\(-9x^4\)[/tex], factoring out [tex]\(-9x^2\)[/tex] leaves us with [tex]\(x^2\)[/tex].
- From [tex]\(45x^3\)[/tex], factoring out [tex]\(-9x^2\)[/tex] leaves us with [tex]\(-5x\)[/tex].
- From [tex]\(18x^2\)[/tex], factoring out [tex]\(-9x^2\)[/tex] leaves us with [tex]\(-2\)[/tex].
After factoring out [tex]\(-9x^2\)[/tex], the polynomial becomes:
[tex]\[
-9x^2(x^2 - 5x - 2)
\][/tex]
3. Check the factored expression:
We can verify by distributing [tex]\(-9x^2\)[/tex] back through the polynomial [tex]\((x^2 - 5x - 2)\)[/tex] to ensure it simplifies back to the original polynomial:
[tex]\[
-9x^2 \cdot x^2 = -9x^4, \quad
-9x^2 \cdot (-5x) = 45x^3, \quad
-9x^2 \cdot (-2) = 18x^2
\][/tex]
The expanded form matches the original polynomial, confirming our factorization is correct.
So, the factorized form of [tex]\(-9x^4 + 45x^3 + 18x^2\)[/tex] using the negative of the greatest common factor is:
[tex]\[
-9x^2(x^2 - 5x - 2)
\][/tex]
