# Difference between revisions of "004 Sample Final A, Problem 8"

a) List all the possible rational zeros of the function ${\displaystyle f(x)=x^{4}-4x^{3}-7x^{2}+34x-24.}$
b) Find all the zeros, that is, solve ${\displaystyle f(x)=0}$
Foundations
If ${\displaystyle f(x)=x^{4}+bx^{3}+cx^{2}+dx+e}$, what does the rational roots tell us are the possible roots of ${\displaystyle f(x)}$?
The rational roots tells us that the possible roots of ${\displaystyle f(x)}$ are ${\displaystyle \pm k}$ where ${\displaystyle k}$ is a divisor of ${\displaystyle e}$.

Solution:

Step 1:
By the rational roots test, the possible roots of ${\displaystyle f(x)}$ are ${\displaystyle \pm \{1,2,3,4,6,8,12,24\}}$.
Step 2:
Using synthetic division, we test 1 as a root of ${\displaystyle f(x)}$. We get a remainder of 0. So, we have that 1 is a root of ${\displaystyle f(x)}$.
By synthetic division, ${\displaystyle f(x)=(x-1)(x^{3}-3x^{2}-10x+24)}$.
Step 3:
Using synthetic division on ${\displaystyle x^{3}-3x^{2}-10x+24}$, we test 2 as a root of this function. We get a remainder of 0. So, we have that 2 is a root of ${\displaystyle x^{3}-3x^{2}-10x+24}$.
By synthetic division, ${\displaystyle x^{3}-3x^{2}-10x+24=(x-2)(x^{2}-x-12)}$.
Step 4:
Thus, ${\displaystyle f(x)=(x-1)(x-2)(x^{2}-x-12)=(x-1)(x-2)(x-4)(x+3)}$.
The zeros of ${\displaystyle f(x)}$ are ${\displaystyle 1,2,4,-3}$.
The possible roots of ${\displaystyle f(x)}$ are ${\displaystyle \pm \{1,2,3,4,6,8,12,24\}}$.
The zeros of ${\displaystyle f(x)}$ are ${\displaystyle 1,2,4,-3}$