Answer :
We start with the expression:
[tex]$$
3x^8 - 9x^7 + 18x^6.
$$[/tex]
Step 1. Identify the Greatest Common Factor (GCF):
- Coefficients: The coefficients are 3, -9, and 18. The GCF of these numbers is 3.
- Variable Part: The terms have exponents 8, 7, and 6 on [tex]$x$[/tex]. The smallest exponent is 6, so we have a common factor of [tex]$x^6$[/tex].
Thus, the GCF for the entire expression is [tex]$3x^6$[/tex].
Step 2. Factor Out the GCF:
Factor [tex]$3x^6$[/tex] out of each term:
- For [tex]$3x^8$[/tex]:
[tex]$$
3x^8 = 3x^6 \cdot x^2.
$$[/tex]
- For [tex]$-9x^7$[/tex]:
[tex]$$
-9x^7 = 3x^6 \cdot (-3x).
$$[/tex]
- For [tex]$18x^6$[/tex]:
[tex]$$
18x^6 = 3x^6 \cdot 6.
$$[/tex]
Step 3. Write the Expression in Factored Form:
After factoring out the GCF, the expression becomes:
[tex]$$
3x^6(x^2 - 3x + 6).
$$[/tex]
Therefore, the simplified factored form of the expression is:
[tex]$$
3x^6(x^2 - 3x + 6).
$$[/tex]
[tex]$$
3x^8 - 9x^7 + 18x^6.
$$[/tex]
Step 1. Identify the Greatest Common Factor (GCF):
- Coefficients: The coefficients are 3, -9, and 18. The GCF of these numbers is 3.
- Variable Part: The terms have exponents 8, 7, and 6 on [tex]$x$[/tex]. The smallest exponent is 6, so we have a common factor of [tex]$x^6$[/tex].
Thus, the GCF for the entire expression is [tex]$3x^6$[/tex].
Step 2. Factor Out the GCF:
Factor [tex]$3x^6$[/tex] out of each term:
- For [tex]$3x^8$[/tex]:
[tex]$$
3x^8 = 3x^6 \cdot x^2.
$$[/tex]
- For [tex]$-9x^7$[/tex]:
[tex]$$
-9x^7 = 3x^6 \cdot (-3x).
$$[/tex]
- For [tex]$18x^6$[/tex]:
[tex]$$
18x^6 = 3x^6 \cdot 6.
$$[/tex]
Step 3. Write the Expression in Factored Form:
After factoring out the GCF, the expression becomes:
[tex]$$
3x^6(x^2 - 3x + 6).
$$[/tex]
Therefore, the simplified factored form of the expression is:
[tex]$$
3x^6(x^2 - 3x + 6).
$$[/tex]
