005 Sample Final A, Question 6

Question Factor the following polynomial completely,     $p(x)=x^{4}+x^{3}+2x-4$

Foundations
1) What does the Rational Zeros Theorem say about the possible zeros?
2) How do you check if a possible zero is actually a zero?
3) How do you find the rest of the zeros?
Answer:
1) The possible divisors can be found by finding the factors of -10, in a list, and the factors of 1, in a second list. Then write down all the fractions with numerators from the first list and denominators from the second list.
2) Use synthetic division, or plug a possible zero into the function. If you get 0, you have found a zero.
3) After your reduce the polynomial with synthetic division, try and find another zero from the list you made in part a). Once you reach a degree 2 polynomial you can finish the problem with the quadratic formula.

Step 1:
First, we use the Rational Zeros Theorem to note that the possible zeros are: $\{\pm 1,\pm 2,\pm 4\}$

Step 2:
Now we start checking which of the possible roots are actually roots. We find that 1 is a zero, and apply either synthetic division or long division to get $x^{4}+x^{3}+2x-4=(x-1)(x^{3}+2x^{2}+2x+4)$

Step 3:
We continue checking zeros and find that -2 is a zero. Applying synthetic division or long division we can simplify the polynomial down to:
$x^{4}+x^{3}+2x-4=(x-1)(x+2)(x^{2}+2)$

Step 4:
Now we can finish the problem by applying the quadratic formula or just finding the roots of $x^{2}+2$

Final Answer:
$x^{4}+x^{3}+2x-4=(x-1)(x+2)(x-{\sqrt {2}}i)(x+{\sqrt {2}}i)$