A binomial is a factor of a polynomial if the polynomial can be written as a product of the binomial and another polynomial.
See solution.
Practice makes perfect
We want to examine how the division of a polynomial by a binomial can be used to determine if the binomial is a factor of the polynomial. Suppose we divide a polynomial by a binomial. The result of any division is the quotient plus the remainder, so we can write the result of our division in the following form.
polynomial/binomial=quotient+remainder/binomial
Since we are dividing the polynomial by the binomial, both the quotient and the remainder are also polynomials. Let's multiply both sides of the above equation by the binomial.
The result of the division — the quotient and the remainder — can be used to express the original polynomial. If the remainder is 0, we are able to write the polynomial as the product of the binomial and the quotient, which is just another polynomial.
remainder= 0
⇓
polynomial= binomial*quotient + 0
Now let's look at it from the opposite direction. If we can write the original polynomial as the product of the binomial and another polynomial, the remainder will be 0.
polynomial= binomial*another polynomial
⇓
polynomial/binomial=another polynomial+ 0
We can say that a binomial is a factor of a polynomial if and only if the polynomial can be written as a product of the binomial and another polynomial. Therefore, if the remainder is equal to 0, the binomial is a factor of the polynomial.
