Answer :
Sure, let's factor out the greatest common factor from the given polynomial [tex]\( 9x^7 - 81x^6 + 36x^5 \)[/tex].
Step-by-step solution:
1. Identify the terms:
The polynomial is made up of three terms: [tex]\( 9x^7 \)[/tex], [tex]\( -81x^6 \)[/tex], and [tex]\( 36x^5 \)[/tex].
2. Find the greatest common factor (GCF) of the coefficients:
- The coefficients are 9, -81, and 36.
- The greatest common factor of these coefficients is 9.
3. Find the smallest power of [tex]\( x \)[/tex] among the terms:
- The powers of [tex]\( x \)[/tex] are 7, 6, and 5.
- The smallest power is [tex]\( x^5 \)[/tex].
4. Combine the GCF of the coefficients and the smallest power of [tex]\( x \)[/tex]:
- The GCF of the entire polynomial is [tex]\( 9x^5 \)[/tex].
5. Factor out the GCF from each term:
- Write each term as a product of the GCF and another term:
[tex]\[
9x^7 = (9x^5) \cdot x^2
\][/tex]
[tex]\[
-81x^6 = (9x^5) \cdot (-9x)
\][/tex]
[tex]\[
36x^5 = (9x^5) \cdot 4
\][/tex]
6. Express the polynomial as the product of the GCF and the remaining polynomial:
- The original polynomial [tex]\( 9x^7 - 81x^6 + 36x^5 \)[/tex] can be written as:
[tex]\[
9x^5 (x^2 - 9x + 4)
\][/tex]
Therefore, the polynomial [tex]\( 9x^7 - 81x^6 + 36x^5 \)[/tex] factored out with the greatest common factor is:
[tex]\[
9x^5 (x^2 - 9x + 4)
\][/tex]
Step-by-step solution:
1. Identify the terms:
The polynomial is made up of three terms: [tex]\( 9x^7 \)[/tex], [tex]\( -81x^6 \)[/tex], and [tex]\( 36x^5 \)[/tex].
2. Find the greatest common factor (GCF) of the coefficients:
- The coefficients are 9, -81, and 36.
- The greatest common factor of these coefficients is 9.
3. Find the smallest power of [tex]\( x \)[/tex] among the terms:
- The powers of [tex]\( x \)[/tex] are 7, 6, and 5.
- The smallest power is [tex]\( x^5 \)[/tex].
4. Combine the GCF of the coefficients and the smallest power of [tex]\( x \)[/tex]:
- The GCF of the entire polynomial is [tex]\( 9x^5 \)[/tex].
5. Factor out the GCF from each term:
- Write each term as a product of the GCF and another term:
[tex]\[
9x^7 = (9x^5) \cdot x^2
\][/tex]
[tex]\[
-81x^6 = (9x^5) \cdot (-9x)
\][/tex]
[tex]\[
36x^5 = (9x^5) \cdot 4
\][/tex]
6. Express the polynomial as the product of the GCF and the remaining polynomial:
- The original polynomial [tex]\( 9x^7 - 81x^6 + 36x^5 \)[/tex] can be written as:
[tex]\[
9x^5 (x^2 - 9x + 4)
\][/tex]
Therefore, the polynomial [tex]\( 9x^7 - 81x^6 + 36x^5 \)[/tex] factored out with the greatest common factor is:
[tex]\[
9x^5 (x^2 - 9x + 4)
\][/tex]
