Question

1. factor (9u^{3}v^{3} + 90u^{6}v^{4}-18u^{5}v^{2}).

a. (3u^{2}v(3u^{4}v^{3}+30u^{4}v^{4}-6u^{3}v^{2}))

b. (9u^{3}v^{2}(v + 10u^{3}v^{2}-2u^{2}))

c. (u^{3}v^{2}(9u + 90u^{3}v^{2}-18u^{2}))

d. (9u^{3}v^{2}(u^{4}v + 10uv^{2}-2))

Explanation:

Step1: Identify GCF of coefficients

The coefficients are 9, 90, -18.
GCF of 9,90,18 is $9$.

Step2: Identify GCF of $u$-terms

The $u$-terms are $u^3, u^6, u^5$.
GCF is $u^{\min(3,6,5)} = u^2$.

Step3: Identify GCF of $v$-terms

The $v$-terms are $v^3, v^4, v^2$.
GCF is $v^{\min(3,4,2)} = v^2$.

Step4: Factor out overall GCF

Overall GCF is $9u^2v^2$. Divide each term:
$\frac{9u^3v^3}{9u^2v^2}=u^4v$, $\frac{90u^6v^4}{9u^2v^2}=10uv^2$, $\frac{-18u^5v^2}{9u^2v^2}=-2$.
Result: $9u^2v^2(u^4v + 10uv^2 - 2)$

Answer:

d. $9u^{2}v^{2}(u^{4}v + 10uv^{2} - 2)$