# Thread: Need help with Factoring and Solving Equations #3.

1. ## Need help with Factoring and Solving Equations #3.

Hi! Ive been trying to do these problems for a while and Im stumped. I finished 13 problems and fully understand them , but I don't know how to do these problems. It would be nice if you could write each step out or the answer at the very least. Or hints or tips, like which theorem or something to use. Shortcuts.

6. Find all the roots for
2x^4 + 11x^3 + 17x^2 - 5x - 25 = 0
For this, how do you start it. Like what is the most efficient way. Also is there any shortcuts?

7. f(x) = 2x + x^2 - x + 3k
f(x) / x+2k has a remainder of -5, find k

2. Then for number 7.
7. f(x) = 2x + x^2 - x + 3k
f(x) / x+2k has a remainder of -5, find k
What do i plug in? because it has k in the x+2k.
i plug in x-2k? and then set it equal to -5?

3. By the remainder theorem, the remainder of $\displaystyle \frac{f(x)}{x - a}$ is $\displaystyle f(a)$.

So if you're dividing by $\displaystyle x - 2k$, the remainder is $\displaystyle f(2k)$.

So you are going to solve for $\displaystyle k$ with $\displaystyle f(2k) = -5$.

4. Originally Posted by donaldmax
6. Find all the roots for
2x^4 + 11x^3 + 17x^2 - 5x - 25 = 0
For this, how do you start it. Like what is the most efficient way. Also is there any shortcuts?
no "shortcuts" for this one ... use the rational root theorem for possible rational roots.
Use of the theorem let me find that x = 1 is a root.

5. Originally Posted by skeeter
no "shortcuts" for this one ... use the rational root theorem for possible rational roots.
Use of the theorem let me find that x = 1 is a root.
Actually there's one "almost" shortcut. Substituting a 1 for x is the same as adding all the coefficients. So if all the coefficients add to 0, then x=1 is a root.