Answer :
To write an equivalent expression for [tex]\(15x^6 + 45x^3 + 30x^4\)[/tex] using the greatest common monomial factor, we need to follow these steps:
1. Identify the Greatest Common Factor (GCF) of the Coefficients:
The coefficients of the terms in the expression are 15, 45, and 30. The GCF of these numbers is 15.
2. Identify the Greatest Common Factor of the Variables:
The variable parts of the terms are [tex]\(x^6\)[/tex], [tex]\(x^3\)[/tex], and [tex]\(x^4\)[/tex].
To find the GCF, look for the lowest power of [tex]\(x\)[/tex] present in all terms, which is [tex]\(x^3\)[/tex].
3. Form the Greatest Common Monomial Factor:
Combine the GCF of the coefficients (15) and the GCF of the variables ([tex]\(x^3\)[/tex]), the greatest common monomial factor is [tex]\(15x^3\)[/tex].
4. Factor Out the Greatest Common Monomial Factor:
Divide each term in the original expression by the greatest common monomial factor and write the expression as a product:
- [tex]\( \frac{15x^6}{15x^3} = x^3 \)[/tex]
- [tex]\( \frac{45x^3}{15x^3} = 3 \)[/tex]
- [tex]\( \frac{30x^4}{15x^3} = 2x \)[/tex]
So, the factored expression is:
[tex]\[
15x^3(x^3 + 3 + 2x)
\][/tex]
Therefore, the equivalent expression is [tex]\(15x^3(x^3 + 3 + 2x)\)[/tex].
1. Identify the Greatest Common Factor (GCF) of the Coefficients:
The coefficients of the terms in the expression are 15, 45, and 30. The GCF of these numbers is 15.
2. Identify the Greatest Common Factor of the Variables:
The variable parts of the terms are [tex]\(x^6\)[/tex], [tex]\(x^3\)[/tex], and [tex]\(x^4\)[/tex].
To find the GCF, look for the lowest power of [tex]\(x\)[/tex] present in all terms, which is [tex]\(x^3\)[/tex].
3. Form the Greatest Common Monomial Factor:
Combine the GCF of the coefficients (15) and the GCF of the variables ([tex]\(x^3\)[/tex]), the greatest common monomial factor is [tex]\(15x^3\)[/tex].
4. Factor Out the Greatest Common Monomial Factor:
Divide each term in the original expression by the greatest common monomial factor and write the expression as a product:
- [tex]\( \frac{15x^6}{15x^3} = x^3 \)[/tex]
- [tex]\( \frac{45x^3}{15x^3} = 3 \)[/tex]
- [tex]\( \frac{30x^4}{15x^3} = 2x \)[/tex]
So, the factored expression is:
[tex]\[
15x^3(x^3 + 3 + 2x)
\][/tex]
Therefore, the equivalent expression is [tex]\(15x^3(x^3 + 3 + 2x)\)[/tex].
