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Multiplying a Polynomial By a Monomial

Multiplying a Polynomial By a Monomial

We have used the Distributive Property to simplify expressions like \(2\left(x-3\right)\). You multiplied both terms in the parentheses, \(x\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}3\), by 2, to get \(2x-6\). With this tutorial’s new vocabulary, you can say you were multiplying a binomial, \(x-3\), by a monomial, 2.

Multiplying a binomial by a monomial is nothing new for you! Here’s an example:

Example

Multiply: \(4\left(x+3\right).\)

Solution

4 times x plus 3. Two arrows extend from 4, terminating at x and 3.
Distribute. 4 times x plus 4 times 3.
Simplify. 4 x plus 12.

Example

Multiply: \(y\left(y-2\right).\)

Solution

y times y minus 2. Two arrows extend from the coefficient y, terminating at the y and minus 2 in parentheses.
Distribute. y times y minus y times 2.
Simplify. y squared minus 2 y.

Example

Multiply: \(7x\left(2x+y\right).\)

Solution

7 x times 2 x plus y. Two arrows extend from 7x, terminating at 2x and y.
Distribute. 7 x times 2 x plus 7 x times y.
Simplify. 14 x squared plus 7 x y.

Example

Multiply: \(-2y\left(4{y}^{2}+3y-5\right).\)

Solution

Negative 2 y times 4 y squared plus 3 y minus 5. Three arrows extend from negative 2 y, terminating at 4 y squared, 3 y, and minus 5.
Distribute. Negative 2 y times 4 y squared plus negative 2 y times 3 y minus negative 2 y times 5.
Simplify. Negative 8 y cubed minus 6 y squared plus 10 y.

Example

Multiply: \(2{x}^{3}\left({x}^{2}-8x+1\right).\)

Solution

2 x cubed times x squared minus 8 x plus 1. Three arrows extend from 2 x cubed, terminating at x squared, minus 8 x, and 1.
Distribute. 2 x cubed times x squared plus 2 x cubed times negative 8 x plus 2 x cubed times 1.
Simplify. 2 x to the fifth power minus 16 x to the fourth power plus 2 x cubed.

Example

Multiply: \(\left(x+3\right)p.\)

Solution

The monomial is the second factor. x plus 3, in parentheses, times p. Two arrows extend from the p, terminating at x and 3.
Distribute. x times p plus 3 times p.
Simplify. x p plus 3 p.

This lesson is part of:

Polynomials II

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