Long division with polynomials

Question

(a^5 + 6a^4 - 2a^2 + 9a + 18) divided by (a^2 - 2)

then

use the factor theorem to determine if x - 1 is a factor of p(x) = x^4 + 4x^3 - 7x^2 - 22x + 24

Answer 1

Hi there! I would love to help you out!

Before setting the first problem up, you need to add a few terms. In the first expression, you'll notice we are missing a term for "a^3", so we will need to add an "0a^3". In the second expression, we are missing an "a" so we will add "0a". This now gives us:

(a^5 + 6a^4 + 0a^3 - 2a^2 + 9a + 18) divided by (a^2 + 0a - 2)

Below is a picture of the long division problem worked out:

Our final answer from the division problem is a³+6a²+2a+10+[(13a+38)/(a²-2)]. Explaining the work done on a long division problem with polynomials is tricky, so if you have questions about the work presented, please don't hesitate to reach out.

For your second problem, we need to figure out if (x - 1) is a factor of x^4+4x^3-7x^2-22x+24. We can do this using synthetic division. We are going to synthetically divide 1 into our polynomial. If we are left with a remainder, that means that (x - 1) is not a factor. Below is the work for the synthetic division:

You can see we are left with a remainder of 0 (which means there is no remainder!). Since there is no remainder, we know that (x - 1) is a factor of our polynomial.

Again, I realize it's difficult to explain long and synthetic division in word format so if you have any specific questions about these steps please don't hesitate to reach out. Learn on! :)