need assistance w/ factoring x^4 + 13x^2 - 4x^3 - 36x + 36

[500919]

here is the problem....

Factor x^4 + 13x^2 - 4x^3 - 36x + 36

 

[tkhunny]

Have you tried the Rational Root Theorem?

Try 1, 2, 3, 4, 6. Synthetic Division?

1) Why did I quit at 6, rather than suggesting also 9, 12, 18, and 36?
2) How do you turn Roots into Factors?

 

[soroban]

Re: need assistance w/ factoring x^4 + 13x^2 - 4x^3 - 36x +

Hello, 500919!

I found the factors . . . with a bit of luck.

Factor: \(\displaystyle \:x^4\,+\,13x^2\,-\,4x^3\,-\,36x\,+\,36\)

Rearrange the terms; \(\displaystyle \x^4\,+\,13x^2\,+\,36)\,-\,4x^3\,-\,36x\)

Factor "by grouping": \(\displaystyle \x^2\,+\,4)(x^2\,+\,9)\,-\,4x(x^2\,+\,9)\)

Factor out the common factor: \(\displaystyle \x^2\,+\,9)\,(x^2\,+\,4\,-\,4x)\)

. . and we have: \(\displaystyle \x^2\,+\,9)(x^2\,-\,4x\,+\,4)\)

. . which factors: \(\displaystyle \x^2\,+\,9)(x\,-\,2)^2\;\;\) . . .ta-DAA!

 

[stapel]

I found the factors . . . \(\displaystyle \x^2\,+\,9)(x\,-\,2)^2\;\;\) . . .ta-DAA!

It's unfortunate that the student was not allowed the opportunity to learn from the hints provided earlier, invest him/herself in the process, and reply with what s/he'd done. How sad.

Eliz.