004 Sample Final A, Problem 8

a) List all the possible rational zeros of the function $f(x)=x^{4}-4x^{3}-7x^{2}+34x-24.$

b) Find all the zeros, that is, solve $f(x)=0$

Foundations
If $f(x)=x^{4}+bx^{3}+cx^{2}+dx+e$, what does the rational roots tell us are the possible roots of $f(x)$?
Answer:
The rational roots tells us that the possible roots of $f(x)$ are $\pm k$ where $k$ is a divisor of $e$.

Solution:

Step 1:
By the rational roots test, the possible roots of $f(x)$ are $\pm \{1,2,3,4,6,8,12,24\}$.

Step 2:
Using synthetic division, we test 1 as a root of $f(x)$. We get a remainder of 0. So, we have that 1 is a root of $f(x)$.
By synthetic division, $f(x)=(x-1)(x^{3}-3x^{2}-10x+24)$.

Step 3:
Using synthetic division on $x^{3}-3x^{2}-10x+24$, we test 2 as a root of this function. We get a remainder of 0. So, we have that 2 is a root of $x^{3}-3x^{2}-10x+24$.
By synthetic division, $x^{3}-3x^{2}-10x+24=(x-2)(x^{2}-x-12)$.

Step 4:
Thus, $f(x)=(x-1)(x-2)(x^{2}-x-12)=(x-1)(x-2)(x-4)(x+3)$.
The zeros of $f(x)$ are $1,2,4,-3$.

Final Answer:
The possible roots of $f(x)$ are $\pm \{1,2,3,4,6,8,12,24\}$.
The zeros of $f(x)$ are $1,2,4,-3$

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