Question

find the product:$(a^{2})(2a^{3})(a^{2}-6a + 9)LXB02a^{7}-16a^{6}-18a^{5}LXB12a^{12}-18a^{7}+18a^{5}$completeconsider the degree of each polynomial in the problem.the first factor has a degree of $\boldsymbol{2}$the second factor has a degree of $\boldsymbol{3}$the third factor has a degree of $\boldsymbol{2}$the product has a degree of $\boldsymbol{7}$completethe degree of the product is the of the degrees of the factors.done

Explanation:

Step1: Simplify initial factors

First, multiply the first two monomials:
$a^2 \times 2a^3 = 2a^{2+3} = 2a^5$

Step2: Distribute to trinomial

Multiply $2a^5$ by each term in $a^2 - 8a + 9$:
$2a^5 \times a^2 = 2a^{5+2} = 2a^7$
$2a^5 \times (-8a) = -16a^{5+1} = -16a^6$
$2a^5 \times 9 = 18a^5$

Step3: Sum the terms

Combine the results from Step2:
$2a^7 - 16a^6 + 18a^5$

Step4: Calculate factor degrees

• Degree of $a^2$: $2$
• Degree of $2a^3$: $3$
• Degree of $a^2 - 8a + 9$: $2$

Step5: Calculate product degree

Add the degrees of the factors:
$2 + 3 + 2 = 7$

Step6: Fill the final blank

The degree of a product is the sum of the degrees of its factors.

Answer:

1. Product: $\boldsymbol{2a^7 - 16a^6 + 18a^5}$
2. First factor degree: $\boldsymbol{2}$
3. Second factor degree: $\boldsymbol{3}$
4. Third factor degree: $\boldsymbol{2}$
5. Product degree: $\boldsymbol{7}$
6. Final blank: $\boldsymbol{sum}$