Question

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

a. ( 9u^3v^2(v + 10u^3v^2 - 2u^2) )
b. ( 3u^2v(3u^4v^3 + 30u^4v^4 - 6u^3v^2) )
c. ( u^3v^2(9u + 90u^3v^2 - 18u^2) )
d. ( 9u^2v^2(u^4v + 10uw^2 - 2) )

Explanation:

Step1: Identify GCF of terms

Find GCF of $9u^3v^3$, $90u^6v^4$, $18u^5v^2$:
GCF of coefficients: $\gcd(9,90,18)=9$
GCF of $u$-terms: $u^{\min(3,6,5)}=u^3$
GCF of $v$-terms: $v^{\min(3,4,2)}=v^2$
Overall GCF: $9u^3v^2$

Step2: Divide each term by GCF

$\frac{9u^3v^3}{9u^3v^2}=v$
$\frac{90u^6v^4}{9u^3v^2}=10u^3v^2$
$\frac{-18u^5v^2}{9u^3v^2}=-2u^2$

Step3: Write factored form

$9u^3v^2(v + 10u^3v^2 - 2u^2)$

Answer:

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