Dividing a Polynomial by a Monomial

0:00How do we divide 15 k squared minus 10 k by 5 k?

0:05This is a polynomial, a sum of terms. It has two terms: 15 k squared and minus 10 k.

0:145 k is a monomial: a polynomial with just one term. We take k as not zero, since we cannot divide by zero.

0:24Picture a rectangle with area 15 k squared minus 10 k and height 5 k.

0:31Area equals height times width, so the width is the area divided by the height.

0:38Split the rectangle into one part for each term: 15 k squared and minus 10 k.

0:45Divide the first term by 5 k. Numbers first: 15 divided by 5 is 3.

0:53Then the letters: k squared is k times k, so k squared divided by k is k.

1:01So 15 k squared divided by 5 k is 3 k.

1:06Now the second term. Minus 10 divided by 5 is minus 2, and k divided by k is 1, so no k is left.

1:16So minus 10 k divided by 5 k is minus 2.

1:22So the width is 3 k minus 2, and that is the answer to our division.

1:28Check by multiplying back. 5 k times 3 k is 15 k squared, and 5 k times minus 2 is minus 10 k.

1:39Together that gives 15 k squared minus 10 k, the polynomial we started with.

1:45A common mistake is to divide only the first term, giving 3 k minus 10 k. The minus 10 k was never divided.

1:55Multiplying back gives 15 k squared minus 50 k squared, not 15 k squared minus 10 k.

2:03To divide a polynomial by a monomial, divide every term by the monomial.