Answer :
We are given the polynomial
[tex]$$8x^6 + 12x^4 + 20x^3.$$[/tex]
Step 1: Find the greatest common factor (GCF) of the coefficients
The coefficients are 8, 12, and 20. The greatest common factor of these numbers is 4.
Step 2: Find the greatest common factor from the variable terms
The exponents of [tex]\( x \)[/tex] in the polynomial are 6, 4, and 3. The smallest exponent is 3, so [tex]\( x^3 \)[/tex] is common to all terms.
Step 3: Factor out the GCF
The overall GCF of the polynomial is [tex]\( 4x^3 \)[/tex]. Now, we factor [tex]\( 4x^3 \)[/tex] from each term:
- For [tex]\( 8x^6 \)[/tex]:
[tex]\[
\frac{8x^6}{4x^3} = 2x^{6-3} = 2x^3;
\][/tex]
- For [tex]\( 12x^4 \)[/tex]:
[tex]\[
\frac{12x^4}{4x^3} = 3x^{4-3} = 3x;
\][/tex]
- For [tex]\( 20x^3 \)[/tex]:
[tex]\[
\frac{20x^3}{4x^3} = 5x^{3-3} = 5.
\][/tex]
Step 4: Write the final factored form
After factoring out [tex]\( 4x^3 \)[/tex], the expression inside the parentheses is [tex]\( 2x^3 + 3x + 5 \)[/tex]. Therefore, the polynomial can be written as:
[tex]$$4x^3 \left( 2x^3 + 3x + 5 \right).$$[/tex]
[tex]$$8x^6 + 12x^4 + 20x^3.$$[/tex]
Step 1: Find the greatest common factor (GCF) of the coefficients
The coefficients are 8, 12, and 20. The greatest common factor of these numbers is 4.
Step 2: Find the greatest common factor from the variable terms
The exponents of [tex]\( x \)[/tex] in the polynomial are 6, 4, and 3. The smallest exponent is 3, so [tex]\( x^3 \)[/tex] is common to all terms.
Step 3: Factor out the GCF
The overall GCF of the polynomial is [tex]\( 4x^3 \)[/tex]. Now, we factor [tex]\( 4x^3 \)[/tex] from each term:
- For [tex]\( 8x^6 \)[/tex]:
[tex]\[
\frac{8x^6}{4x^3} = 2x^{6-3} = 2x^3;
\][/tex]
- For [tex]\( 12x^4 \)[/tex]:
[tex]\[
\frac{12x^4}{4x^3} = 3x^{4-3} = 3x;
\][/tex]
- For [tex]\( 20x^3 \)[/tex]:
[tex]\[
\frac{20x^3}{4x^3} = 5x^{3-3} = 5.
\][/tex]
Step 4: Write the final factored form
After factoring out [tex]\( 4x^3 \)[/tex], the expression inside the parentheses is [tex]\( 2x^3 + 3x + 5 \)[/tex]. Therefore, the polynomial can be written as:
[tex]$$4x^3 \left( 2x^3 + 3x + 5 \right).$$[/tex]
