Question

Suppose you divide a polynomial by a binomial. How do you know if the binomial is a factor of the polynomial? Create a example problem that has a binomial which IS a factor of the polynomial being divided, and another problem that has a binomial which is NOT a factor of the polynomial being divided.

Answer 1

The binomial is a factor if the remainder is zero after division.

Answer 2

Remainder theorem:
\[\frac{p(x)}{f(x)}=q(x)+r(x)\]
i.e. a polynomial divided by another function equals a quotient function plus a remainder function. If the remainder equals zero, then f(x) is a factor of p(x).
\[\rightarrow p(x)=q(x)f(x)\]

Answer 3

huh

Answer 4

pick whatever functions you want for q(x), f(x), and r(x), then work backwards to get the polynomial dividend, p(x).

Answer 5

Here's an example from arithmetic: \[108\div9=12\]
9 is a factor of 108, so there is no remainder.
Here is the same statement using algebra:
\[(x^2+x-2) \div (x-1)=(x+2)\]
Let x=10 in the above, and you'll see it's exactly the same. :-)