Adding, subtracting and multiplying polynomials

Adding and subtracting

To add polynomials, write all the terms in a row and combine like terms. The only trap is subtraction.

A monomial times a polynomial

The distributive property does all the work: the monomial multiplies each term inside the parentheses.

A polynomial times a polynomial

If the first polynomial has \(m\) terms and the second has \(n\), you get \(m \cdot n\) products before simplifying. Counting them is a good way to make sure you haven't skipped one.

For bigger products, a grid (often called the box method) keeps everything organized: one polynomial labels the rows, the other the columns, and each cell holds a product.

Checking a product

Here's a fast check that catches most mistakes: plug the same easy number into both sides. If they disagree, there's an error.

With \(a = 1\) in the last example: the left side gives \((3 + 2)(1 - 4 + 1) = 5 \cdot (-2) = -10\), and the right side gives \(3 - 10 - 5 + 2 = -10\). They match.

Several factors

Multiply two at a time.

Practice