Answer :
We start with the expression
[tex]$$21x^7 + 9x^6 + 33x^4.$$[/tex]
Step 1. Find the Greatest Common Factor (GCF) of the Numeric Coefficients
The coefficients are 21, 9, and 33. Their greatest common factor is 3.
Step 2. Find the GCF of the Variable Parts
The variable [tex]$x$[/tex] appears in each term with exponents 7, 6, and 4. The smallest exponent is 4. So, the common variable factor is [tex]$x^4$[/tex].
Step 3. Write the Overall GCF
The overall GCF of the expression is
[tex]$$3x^4.$$[/tex]
Step 4. Factor Out the GCF
Divide each term in the expression by [tex]$3x^4$[/tex]:
- For [tex]$21x^7$[/tex]:
[tex]$$\frac{21x^7}{3x^4} = 7x^{7-4} = 7x^3.$$[/tex]
- For [tex]$9x^6$[/tex]:
[tex]$$\frac{9x^6}{3x^4} = 3x^{6-4} = 3x^2.$$[/tex]
- For [tex]$33x^4$[/tex]:
[tex]$$\frac{33x^4}{3x^4} = 11.$$[/tex]
Thus, after factoring out [tex]$3x^4$[/tex], we are left with:
[tex]$$7x^3 + 3x^2 + 11.$$[/tex]
Step 5. Write the Fully Factored Expression
The final factored form of the expression is:
[tex]$$3x^4\left(7x^3 + 3x^2 + 11\right).$$[/tex]
[tex]$$21x^7 + 9x^6 + 33x^4.$$[/tex]
Step 1. Find the Greatest Common Factor (GCF) of the Numeric Coefficients
The coefficients are 21, 9, and 33. Their greatest common factor is 3.
Step 2. Find the GCF of the Variable Parts
The variable [tex]$x$[/tex] appears in each term with exponents 7, 6, and 4. The smallest exponent is 4. So, the common variable factor is [tex]$x^4$[/tex].
Step 3. Write the Overall GCF
The overall GCF of the expression is
[tex]$$3x^4.$$[/tex]
Step 4. Factor Out the GCF
Divide each term in the expression by [tex]$3x^4$[/tex]:
- For [tex]$21x^7$[/tex]:
[tex]$$\frac{21x^7}{3x^4} = 7x^{7-4} = 7x^3.$$[/tex]
- For [tex]$9x^6$[/tex]:
[tex]$$\frac{9x^6}{3x^4} = 3x^{6-4} = 3x^2.$$[/tex]
- For [tex]$33x^4$[/tex]:
[tex]$$\frac{33x^4}{3x^4} = 11.$$[/tex]
Thus, after factoring out [tex]$3x^4$[/tex], we are left with:
[tex]$$7x^3 + 3x^2 + 11.$$[/tex]
Step 5. Write the Fully Factored Expression
The final factored form of the expression is:
[tex]$$3x^4\left(7x^3 + 3x^2 + 11\right).$$[/tex]
