Question

How can you put this in factored form? Can you give me step by step, as well, please?

`f(x)=x^4+9x^3+29x^2+81x+180`

Answer 1

`x^4+9x^3+29x^2+81x+180=color(blue)((x+4)(x+5)(x^2+9))`

Explanation:

My method of finding this was not pretty (hopefully someone else, who is more wide awake, can demonstrate a better method.

I used the Rational Factor Theorem to test a series of factors of `180` looking for a zero: `{+-1,+-2,+-3,+-4,+-5,+-6,+-9,...}`
[I actually wrote a snippet of Basic code to do this for me]
until I found `x=-4` was a zero;
this implies that `(x+4)` is a factor.

Dividing the original expression by `(x+4)` gave a quotient of `x^3+5x^2+4x+45`

I could have re-applied the Rational Factor Theorem but instead took a shot at `(x+5)` and found that it worked, giving the factoring:
`color(white)("XXX")(x+4)(x+5)(x^2+9)`
the final factor `(x^2+9)` obviously has no Real factors
so I decided this was the best I could do.