Answer :
To find an equivalent expression for the polynomial [tex]\(15x^6 + 45x^3 + 30x^4\)[/tex] using the greatest common monomial factor, follow these steps:
1. Identify the Greatest Common Factor (GCF):
- Look at the coefficients: 15, 45, and 30.
- The GCF of 15, 45, and 30 is 15.
- Look at the variable parts: [tex]\(x^6\)[/tex], [tex]\(x^3\)[/tex], and [tex]\(x^4\)[/tex].
- The smallest power of [tex]\(x\)[/tex] among the terms is [tex]\(x^3\)[/tex].
So, the greatest common monomial factor of the entire expression is [tex]\(15x^3\)[/tex].
2. Factor out the GCF:
- Divide each term of the polynomial by the GCF, [tex]\(15x^3\)[/tex]:
- [tex]\(15x^6 \div 15x^3 = x^3\)[/tex]
- [tex]\(45x^3 \div 15x^3 = 3\)[/tex]
- [tex]\(30x^4 \div 15x^3 = 2x\)[/tex]
3. Write the factored expression:
- Combine the terms obtained after dividing:
[tex]\[
15x^6 + 45x^3 + 30x^4 = 15x^3(x^3 + 2x + 3)
\][/tex]
Thus, the equivalent expression using the greatest common monomial factor is [tex]\(15x^3(x^3 + 2x + 3)\)[/tex].
1. Identify the Greatest Common Factor (GCF):
- Look at the coefficients: 15, 45, and 30.
- The GCF of 15, 45, and 30 is 15.
- Look at the variable parts: [tex]\(x^6\)[/tex], [tex]\(x^3\)[/tex], and [tex]\(x^4\)[/tex].
- The smallest power of [tex]\(x\)[/tex] among the terms is [tex]\(x^3\)[/tex].
So, the greatest common monomial factor of the entire expression is [tex]\(15x^3\)[/tex].
2. Factor out the GCF:
- Divide each term of the polynomial by the GCF, [tex]\(15x^3\)[/tex]:
- [tex]\(15x^6 \div 15x^3 = x^3\)[/tex]
- [tex]\(45x^3 \div 15x^3 = 3\)[/tex]
- [tex]\(30x^4 \div 15x^3 = 2x\)[/tex]
3. Write the factored expression:
- Combine the terms obtained after dividing:
[tex]\[
15x^6 + 45x^3 + 30x^4 = 15x^3(x^3 + 2x + 3)
\][/tex]
Thus, the equivalent expression using the greatest common monomial factor is [tex]\(15x^3(x^3 + 2x + 3)\)[/tex].
