Earl is building a doghouse whose front is in the shape of a square topped with a triangle. There will be a rectangular door through which the dog can enter and exit the house. Earl wants to find the area of the front of the doghouse so that he can purchase the correct amount of paint. Using the measurements of the front of the house shown below, we can create an expression that combines several variable terms which allows us to solve this problem and others like it.

Measurements of the front of the doghouse Earl is building.

First, find the area of the square in square feet.

Then, find the area of the triangle in square feet.

Next, find the area of the rectangular door in square feet.

Operations on Polynomials

The area of the front of the doghouse can be found by adding the areas of the square and the triangle and then subtracting the area of the rectangle. When we do this, we get [latex]4{x}^{2}+\frac{3}{2}x-x[/latex] ft², or [latex]4{x}^{2}+\frac{1}{2}x[/latex] ft².

In this section, we will examine expressions such as this one, which combine several variable terms.

The area of the front of the doghouse described in the introduction was [latex]4{x}^{2}+\frac{1}{2}x[/latex] ft².

This is an example of a polynomial which is a sum of or difference of terms each consisting of a variable raised to a nonnegative integer power. A number multiplied by a variable raised to an exponent, such as [latex]384\pi[/latex], is known as a coefficient. Coefficients can be positive, negative, or zero, and can be whole numbers, decimals, or fractions. Each product [latex]{a}_{i}{x}^{i}[/latex], such as [latex]384\pi w[/latex], is a term of a polynomial. If a term does not contain a variable, it is called a constant.

A polynomial containing only one term, such as [latex]5{x}^{4}[/latex], is called a monomial. A polynomial containing two terms, such as [latex]2x - 9[/latex], is called a binomial. A polynomial containing three terms, such as [latex]-3{x}^{2}+8x - 7[/latex], is called a trinomial.

We can find the degree of a polynomial by identifying the highest power of the variable that occurs in the polynomial. The term with the highest degree is called the leading term because it is usually written first. The coefficient of the leading term is called the leading coefficient. When a polynomial is written so that the powers are descending, we say that it is in standard form.

Adding and Subtracting Polynomials

We can add and subtract polynomials by combining like terms, which are terms that contain the same variables raised to the same exponents. For example, [latex]5{x}^{2}[/latex] and [latex]-2{x}^{2}[/latex] are like terms and can be added to get [latex]3{x}^{2}[/latex], but [latex]3x[/latex] and [latex]3{x}^{2}[/latex] are not like terms and therefore cannot be added.

Multiplying Polynomials

Multiplying polynomials is a bit more challenging than adding and subtracting polynomials. We must use the distributive property to multiply each term in the first polynomial by each term in the second polynomial. We then combine like terms. We can also use a shortcut called the FOIL method when multiplying binomials. Certain special products follow patterns that we can memorize and use instead of multiplying the polynomials by hand each time. We will look at a variety of ways to multiply polynomials.

Multiplying Polynomials Using the Distributive Property

To multiply a number by a polynomial we use the distributive property. The number must be distributed to each term of the polynomial. In [latex]2\left(x+7\right)[/latex] we can distribute [latex]2[/latex] to obtain the expression [latex]2x+14[/latex]. When multiplying polynomials, the distributive property allows us to multiply each term of the first polynomial by each term of the second. We then add the products together and combine like terms to simplify.

Using FOIL to Multiply Binomials

A shortcut called FOIL is sometimes used to find the product of two binomials. It is called FOIL because we multiply the first terms, the outer terms, the inner terms, and then the last terms of each binomial.

The FOIL method is simply just the distributive property. We are multiplying each term of the first binomial by each term of the second binomial and then combining like terms.

Special Cases of Polynomials

Perfect Square Trinomials

Certain binomial products have special forms. When a binomial is squared, the result is called a perfect square trinomial. We can find the square by multiplying the binomial by itself. However, there is a special form that each of these perfect square trinomials takes, and memorizing the form makes squaring binomials much easier. Let’s look at a few perfect square trinomials to familiarize ourselves with the form.

Notice that the first term of each trinomial is the square of the first term of the binomial and, similarly, the last term of each trinomial is the square of the last term of the binomial. The middle term is double the product of the two terms. Lastly, we see that the first sign of the trinomial is the same as the sign of the binomial.

Difference of Squares

Another special product is called the difference of squares which occurs when we multiply a binomial by another binomial with the same terms but the opposite sign. Let’s see what happens when we multiply [latex]\left(x+1\right)\left(x - 1\right)[/latex] using the FOIL method.

The middle term drops out resulting in a difference of squares. Just as we did with the perfect squares, let’s look at a few examples.

Because the sign changes in the second binomial, the outer and inner terms cancel each other out, and we are left only with the square of the first term minus the square of the last term.

Performing Operations with Polynomials of Several Variables

We have looked at polynomials containing only one variable. However, a polynomial can contain several variables. All of the same rules apply when working with polynomials containing several variables. Consider an example:

Key Equations

Key Concepts

• A polynomial is a sum of terms each consisting of a variable raised to a nonnegative integer power. The degree is the highest power of the variable that occurs in the polynomial. The leading term is the term containing the highest degree, and the leading coefficient is the coefficient of that term.
• We can add and subtract polynomials by combining like terms.
• To multiply polynomials, use the distributive property to multiply each term in the first polynomial by each term in the second. Then add the products.
• FOIL (First, Outer, Inner, Last) is a shortcut that can be used to multiply binomials.
• Perfect square trinomials and difference of squares are special products.
• Follow the same rules to work with polynomials containing several variables.

Glossary

binomial

a polynomial containing two terms

coefficient

any real number [latex]{a}_{i}[/latex] in a polynomial of the form [latex]{a}_{n}{x}^{n}+\dots+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}[/latex]

degree

the highest power of the variable that occurs in a polynomial

difference of squares

the binomial that results when a binomial is multiplied by a binomial with the same terms, but the opposite sign

leading coefficient

the coefficient of the leading term

leading term

 the term containing the highest degree

monomial

a polynomial containing one term

perfect square trinomial

the trinomial that results when a binomial is squared

polynomial

a sum of terms each consisting of a variable raised to a nonnegative integer power

term of a polynomial

any [latex]{a}_{i}{x}^{i}[/latex] of a polynomial of the form [latex]{a}_{n}{x}^{n}+\dots+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}[/latex]

trinomial

a polynomial containing three terms