Answer :
To factor out the greatest common factor (GCF) from the polynomial [tex]\(12x^5 + 16x^6 - 20x^2 + 28\)[/tex], follow these steps:
1. Identify the GCF of the coefficients:
- Look at the coefficients of the terms: 12, 16, -20, and 28.
- Find the greatest common factor of these numbers.
- The GCF of 12, 16, -20, and 28 is 4.
2. Check for common variable factors:
- The GCF for any variable parts must include the variable raised to the lowest power that appears in all the terms.
- The terms with variable [tex]\(x\)[/tex] are [tex]\(12x^5\)[/tex], [tex]\(16x^6\)[/tex], and [tex]\(-20x^2\)[/tex]. The smallest power of [tex]\(x\)[/tex] in these terms is [tex]\(x^2\)[/tex]. However, since the constant term (28) does not have [tex]\(x\)[/tex], there is no common variable factor in all terms.
3. Factor out the GCF from each term:
- Divide each term by the identified GCF (4).
- [tex]\( \frac{12x^5}{4} = 3x^5 \)[/tex]
- [tex]\( \frac{16x^6}{4} = 4x^6 \)[/tex]
- [tex]\( \frac{-20x^2}{4} = -5x^2 \)[/tex]
- [tex]\( \frac{28}{4} = 7 \)[/tex]
4. Write the expression in factored form:
- After factoring out the GCF, your expression is:
[tex]\[
4(4x^6 + 3x^5 - 5x^2 + 7)
\][/tex]
Therefore, the factored form of the polynomial [tex]\(12x^5 + 16x^6 - 20x^2 + 28\)[/tex] is:
[tex]\[
4(4x^6 + 3x^5 - 5x^2 + 7)
\][/tex]
1. Identify the GCF of the coefficients:
- Look at the coefficients of the terms: 12, 16, -20, and 28.
- Find the greatest common factor of these numbers.
- The GCF of 12, 16, -20, and 28 is 4.
2. Check for common variable factors:
- The GCF for any variable parts must include the variable raised to the lowest power that appears in all the terms.
- The terms with variable [tex]\(x\)[/tex] are [tex]\(12x^5\)[/tex], [tex]\(16x^6\)[/tex], and [tex]\(-20x^2\)[/tex]. The smallest power of [tex]\(x\)[/tex] in these terms is [tex]\(x^2\)[/tex]. However, since the constant term (28) does not have [tex]\(x\)[/tex], there is no common variable factor in all terms.
3. Factor out the GCF from each term:
- Divide each term by the identified GCF (4).
- [tex]\( \frac{12x^5}{4} = 3x^5 \)[/tex]
- [tex]\( \frac{16x^6}{4} = 4x^6 \)[/tex]
- [tex]\( \frac{-20x^2}{4} = -5x^2 \)[/tex]
- [tex]\( \frac{28}{4} = 7 \)[/tex]
4. Write the expression in factored form:
- After factoring out the GCF, your expression is:
[tex]\[
4(4x^6 + 3x^5 - 5x^2 + 7)
\][/tex]
Therefore, the factored form of the polynomial [tex]\(12x^5 + 16x^6 - 20x^2 + 28\)[/tex] is:
[tex]\[
4(4x^6 + 3x^5 - 5x^2 + 7)
\][/tex]
