Help!

[joshuambenhase]

I am new to calculus and am having problems and it more so the algebra but maybe more of an understanding. In this problem, i don't understand what to factor:



Could someone help me? Thanks!

 

[soroban]

Hello, joshuambenhase!

This is Algebra 1 stuff . . . and you're taking Calculus?

Factor out the common factor.

\(\displaystyle \frac{dy}{dx} \;=\;(5x-5)^4(-12x^2)(5-x^3)^3 + (5-x^3)^4(20)(5x-5)^3\)

We have: .\(\displaystyle -12x^2(5x-5)^4(5-x^3)^3 + 20(5x-5)^3(5-x^3)^4\)

. . . . . \(\displaystyle =\;-12x^2[5(x-1)]^4(5-x^3)^3 + 20[5(x-1)]^3(5-x^3)^4\)

. . . . . \(\displaystyle =\;-12x^2\cdot 5^4(x-1)^4(5-x^3)^3 + 20\cdot 5^3(x-1)^3(5-x^3)^3\)

. . . . . \(\displaystyle =\;-12x^3\cdot625(x-1)^4(5-x^3)^3 + 20\cdot125(x-1)^3(5-x^3)^4\)

. . . . . \(\displaystyle =\;\underbrace{-7500x^2(x-1)^4(5-x^3)^3} + \underbrace{2500(x-1)^3(5-x^3)^4}\)


We have two terms.

Can you see that both terms have:
. . a factor of 2500
. . a factor of \(\displaystyle (x-1)^3\)
. . a factor of \(\displaystyle (5-x^3)^3\)


Factor them out . . .

\(\displaystyle 2500(x-1)^3(5-x^3)^3\,\left[-3x^2(x-1) + (5-x^3)\right]\)

. . \(\displaystyle =\;2500(x-1)^3(5-x^3)^3(5+3x^2 - 4x^3)\)