Let's factor the polynomial [tex]\(30x^5y^8z - 18xy^4\)[/tex] and determine the correct terms that belong in the parentheses.
1. Identify the Greatest Common Factor (GCF):
- The terms in the polynomial are [tex]\(30x^5y^8z\)[/tex] and [tex]\(18xy^4\)[/tex].
- The numerical coefficients are 30 and 18. The GCF of 30 and 18 is 6.
- For the variable [tex]\(x\)[/tex], the smallest power is [tex]\(x^1\)[/tex].
- For the variable [tex]\(y\)[/tex], the smallest power is [tex]\(y^4\)[/tex].
- The variable [tex]\(z\)[/tex] is present only in one term, so it doesn’t contribute to the GCF.
So, the GCF of [tex]\(30x^5y^8z\)[/tex] and [tex]\(18xy^4\)[/tex] is [tex]\(6xy^4\)[/tex].
2. Factor Out the GCF:
- [tex]\(30x^5y^8z \div 6xy^4 = 5x^4y^4z\)[/tex]
- [tex]\(18xy^4 \div 6xy^4 = 3\)[/tex]
3. Rewrite the Polynomial in Factored Form:
The polynomial [tex]\(30x^5y^8z - 18xy^4\)[/tex] can be factored as:
[tex]\[
30x^5y^8z - 18xy^4 = 6xy^4(5x^4y^4z) - 6xy^4(3)
\][/tex]
4. Represent the Polynomial with Parentheses:
[tex]\[
30x^5y^8z - 18xy^4 = 6xy^4 \left( 5x^4y^4z - 3 \right)
\][/tex]
So, the terms that belong in the parentheses are [tex]\(5x^4y^4z\)[/tex] and 3. Therefore, the factored form of the polynomial is:
[tex]\[
6xy^4 \left( 5x^4y^4z - 3 \right)
\][/tex]
Among the given options, the correct term is [tex]\(5x^4y^4z\)[/tex].
Therefore, the correct terms that belong in the parentheses are:
[tex]\(6 x y^4 \left(5x^4y^4z - 3\right)\)[/tex].
