Adding and Subtracting Binomials

Video Transcript

When adding integers, to simplify the addition, align the numbers by place value, then combine terms along the place value.

Something comparable happens with polynomials. Suppose you’re combining these polynomials. Arrange them as if this were column addition. Instead of place value, you align terms based on the power of the exponents. See how the squared terms are aligned here; the linear terms are aligned here; and the constant terms are aligned here?

Combine like terms. The resulting polynomial is shown here. How do we know the sum is a polynomial? Each of the terms has a constant term with a variable to an integer power. The constant term has an x to the zero power.

Here is another example, in this case two trinomials are combined. Align terms that have the same variable and exponent. There is only one cubic term, so it goes here. There are quadratic terms, so they both go here. There is only one linear term, so it goes here by itself. There are two numerical terms, and they go here. Now combine terms to get the new polynomial.

You can also add polynomials horizontally. In this case, just place common terms in parentheses, as shown here. Notice all the cubic, quadratic, linear, and constant terms are enclosed in parentheses. Combine terms to find the sum.

Like integers, polynomials are closed under addition. Just as the sum of two integers is another integer, the sum of two polynomials is another polynomial.

Let’s now look at polynomial subtraction.

When subtracting integers, to simplify the subtraction, align the numbers by place value, then combine terms along the place value.

Do the same when subtracting polynomials. Arrange them as if this were column subtraction. Instead of place value, you align terms based on the power of the exponents. See how the squared terms are aligned here; the linear terms are aligned here; and the constant terms are aligned here?

You can also subtract polynomials horizontally. As we did with addition, place common terms in parentheses, as shown here. Combine terms to find the difference.

Like integers, polynomials are closed under subtraction. Just as the difference of two integers is another integer, the difference of two polynomials is another polynomial.

Let’s look at some examples of polynomial addition and subtraction.

Each rectangle has side lengths made up of a binomial. The area of each rectangle is a quadratic trinomial, as shown here.

The combined area of the two rectangles is the sum of the two polynomials. The sum is another polynomial.

Let’s look at another example.

This shape has the dimensions shown. Suppose you want to find the area of the white-shaded region. One way to do that is to find the area of the large rectangle and subtract the area of the small square.

This results in the following subtraction expression. The result is a linear binomial.

Let’s look at a final example.

The graph of this roller coaster is based on this polynomial, while the graph of this roller coaster is based on this polynomial.

Watch what happens to the graph of the roller coaster when you subtract the second polynomial from the first. What do you think accounts for the change in the third graph?