11.3.1: Simplifying and Evaluating Polynomials with More than One Term

Learning Objectives

• Evaluate a polynomial for given values of each variable.
• Simplify polynomials by collecting like terms.

Introduction

You have studied polynomials consisting of constants and/or variables combined by addition or subtraction. The variables may include exponents. The examples so far have been limited to expressions such as \(\ 5 x^{4}+3 x^{3}-6 x^{2}+2 x\) containing one variable, but polynomials can also contain multiple variables. An example of a polynomial with two variables is \(\ 4 x^{2} y-2 x y^{2}+x-7\).

Many formulas are polynomials with more than one variable, such as the formula for the surface area of a rectangular prism: \(\ 2 a b+2 b c+2 a c\), where \(\ a\), \(\ b\), and \(\ c\) are the lengths of the three sides. By substituting in the values of the lengths, you can determine the value of the surface area. By applying the same principles for polynomials with one variable, you can evaluate or combine like terms in polynomials with more than one variable.

Evaluating Polynomials for Given Values of Each Variable

When you evaluate an expression for a given value, you substitute that given value in the expression, and find its numerical value. In the following example, \(\ x=-2\), you replace all of the \(\ x^{\prime} s\) with a value of -2 and simplify the expression following the order of operations.

Example

Evaluate \(\ 7 x^{2}-3 x+2 \text { for } x=-2\).

Solution

\(\ 7(-2)^{2}-3(-2)+2\)	Substitute (-2) for each \(\ x\) in the polynomial.
\(\ 7(4)-3(-2)+2\)	Following the order of operations, evaluate exponents first.
\(\ 28+6+2\)	Perform the multiplication next.
\(\ 34+2\)	Combine terms beginning from the left.
\(\ 36\)	Find the sum.

You can follow the same procedure when there are two variables in an expression. Let’s review an example.

Example

Evaluate \(\ 8 c-7 b \text { for } b=4 \text { and } c=5\).

Solution

\(\ 8(5)-7(4)\)	Substitute 5 for each \(\ c\) in the polynomial and 4 for each \(\ b\).
\(\ 40-28\)	Multiply.
\(\ 12\)	Find the difference.

\(\ 12\)

As with polynomials with one variable, you must pay attention to the rules of exponents and the order of operations so that you correctly evaluate an expression with two or more variables.

Example

Evaluate \(\ x^{2}+3 y^{3} \text { for } x=7 \text { and } y=-2\).

Solution

\(\ (7)^{2}+3(-2)^{3}\)	Substitute the given values for \(\ x\) and \(\ y\).
\(\ 49+3(-8)\)	Evaluate the exponents first.
\(\ 49+(-24)\)	Multiply.
\(\ 25\)	Add.

\(\ 25\)

Example

Evaluate \(\ 4 x^{2} y-2 x y^{2}+x-7 \text { for } x=3 \text { and } y=-1\).

Solution

\(\ 4\left(3^{2}\right)(-1)-2(3)(-1)^{2}+3-7\)	Substitute the given values for \(\ x\) and \(\ y\).
\(\ 4(9)(-1)-2(3)(1)+3-7\)	Evaluate the exponents first.
\(\ -36-6+3-7\)	Perform multiplication next.
\(\ \begin{array}{c} -42+3-7 \\ -39-7 \end{array}\)	Perform addition and subtraction from left to right.
\(\ -46\)	Find the difference.

The next example shows how to evaluate a polynomial with two variables. This polynomial is the formula for perimeter of a rectangle.

Example

The formula for the perimeter, \(\ P\), of a rectangle is \(\ 2 a+2 b\) in which \(\ a\) and \(\ b\) are the lengths of the sides of the rectangle.

Evaluate the formula for \(\ a=6\) inches and \(\ b=10\) inches.

Solution

\(\ 2(6)+2(10)\)	Substitute the given values for \(\ a\) and \(\ b\).
\(\ 12+20\)	Multiply.
\(\ 32\)	Add.

Exercise

Evaluate: \(\ 2 x^{3}-x y^{2}+6 \text { for } x=-2 \text { and } y=5\)

1. -158
2. -60
3. 14
4. 40

Answer

1. Incorrect. Unless there are parentheses, an exponent only goes with the variable immediately to its left. So, the exponent 3 does not apply to the 2 in that term, only to the variable \(\ x\). This is true in the next term also; \(\ x\) is not squared. The correct answer is 40.
2. Incorrect. Be more careful with your sign work. The middle term is positive 50, not negative 50. When you substitute in the values of the variables, you get \(\ 2(-2)^{3}-(-2)(5)^{2}+6\), which is simplified to \(\ 2(-8)-(-2)(25)+6\), which equals \(\ -16+50+6\). The correct answer is 40.
3. Incorrect. An exponent indicates the number of times a value is used as a factor. \(\ x^{3}\) means \(\ x\) times \(\ x\) times \(\ x\), not \(\ x\) times 3. \(\ 2(-2)(-2)(-2)-(-2)(5)(5)+6=-16+50+6\). The correct answer is 40.
4. Correct. \(\ 2(-2)^{3}-(-2)(5)^{2}+6=-16+50+6=34+6=40\)

Identifying the Degree of a Polynomial with Two or More Variables

Mathematicians use conventions for writing and describing polynomials. A polynomial with one variable can be described by the number of terms it has and the degree of the term with the greatest exponent. Polynomials are commonly written with their terms in descending order of degree. Let’s start by looking at an example of a polynomial with one variable: \(\ t^{3}-10 t^{2}-5 t-32\). This polynomial has been written in descending order of degree, starting with the term with an exponent of 3 and ending with the term whose degree is 0 because it has no variable. This polynomial is called a third-degree polynomial because its term with the highest degree is the monomial \(\ t^{3}\). (Note that the degree of a monomial, \(\ t^{3}\), is also 3, because the variable \(\ t\) has an exponent of 3.)

When a polynomial has more than one variable, you can still describe it according to its degree and the degree of its terms. It’s a little more complicated. Let’s look at a polynomial with two variables: \(\ 7 x^{2} y-3 x y^{3}+2 x\). This polynomial has three terms and therefore can be called a trinomial. To determine the degree of a term, you find the sum of the exponents of all the variables in the term. Here are some examples:

Terms	Sum of the Exponents	Degree of the Term
\(\ 7 x^{2} y\)	\(\ 2+1=3\)	\(\ 3\)
\(\ -3 x y^{3}\)	\(\ 1+3=4\)	\(\ 4\)
\(\ 2 x\)	\(\ 1=1\)	\(\ 1\)

The degree of a polynomial is the same as the degree of the term with the highest degree. In this case, \(\ 7 x^{2} y-3 x y^{3}+2 x\) is a fourth-degree polynomial.

Exercise

What description below best matches the expression: \(\ 2 x^{4} y-5 x^{3}-10 x y^{3}\)?

1. A twelfth-degree trinomial
2. A fifth-degree trinomial
3. A third-degree polynomial
4. A fourth-degree polynomial

Answer

1. Incorrect. The degree of a polynomial is the same as the degree of the term whose exponents have the greatest sum, not by the sum of all the exponents in the polynomial. The correct answer is a fifth degree trinomial.
2. Correct. \(\ 2 x^{4} y\) has a degree of 5 because the exponents of its variables are 4 and 1, and \(\ 4+1=5\).
3. Incorrect. The degree of a polynomial is not determined by the number of terms it has. The degree is the same as the degree of the term whose exponents have the greatest sum. The correct answer is a fifth degree trinomial.
4. Incorrect. When there are multiple variables, find the degree of a polynomial by determining which term’s exponents have the greatest sum. This is the degree of the polynomial. The \(\ 2 x^{4} y\) term is the term with the highest degree, the sum of the exponents in the term \(\ 2 x^{4} y\) is \(\ 4+1=5\). Since there are 3 terms, this polynomial is called a trinomial. The correct answer is a fifth degree trinomial.

Simplifying Polynomials with More than One Variable by Combining Like Terms

If a polynomial has like terms, the polynomial can be simplified by combining the like terms.

You’ll recall that like terms contain the same variables raised to the same power. If there is more than one variable, the same is true: same exact variable(s) each raised to the same exact power.

The polynomial \(\ 3 x y^{3} z^{2}+5 x y^{3} z^{2}+6 x^{2} y^{3} z\) has like terms that can be combined. \(\ 3 x y^{3} z^{2}\) and \(\ 5 x y^{3} z^{2}\) are like terms because they have the same variables, \(\ x\), \(\ y\), and \(\ z\), raised to the same powers, \(\ x\), \(\ y^{3}\), and \(\ z^{2}\). They can be collected, or combined, to give a result of \(\ 8 x y^{3} z^{2}\). Notice that while \(\ 6 x^{2} y^{3} z\) has the same variables, \(\ x\), \(\ y\), and \(\ z\), the exponents in this term are different, \(\ x^{2}\) instead of \(\ x\), and \(\ z\) instead of \(\ z^{2}\). So, \(\ 6 x^{2} y^{3} z\) cannot be combined with the other terms. Instead, the simplified polynomial is written with two terms: \(\ 8 x y^{3} z^{2}+6 x^{2} y^{3} z\).

Example

Simplify \(\ 2 x y^{2}-8 x-3 x y^{2}+3 x\)

Solution

\(\ 2 x y^{2}-8 x-3 x y^{2}+3 x\)	Identify any like terms.
\(\ \begin{array}{c} 2 x y^{2}-3 x y^{2}-8 x+3 x \\ (2-3) x y^{2}+(-8+3) x \\ -1 x y^{2}+(-5) x \end{array}\)	Combine like terms using the distributive property.
\(\ -1 x y^{2}-5 x\)	Rewrite using subtraction, and check to make sure all like terms have been combined.

\(\ 2 x y^{2}-8 x-3 x y^{2}+3 x=-x y^{2}-5 x\)

As with polynomials with one variable, you can combine like terms in polynomials with more than one variable by combining the coefficients of those like terms and keeping the variable part the same. That step is written in the example below. But to save time, you can also just perform the computation in your head.

Example

Simplify \(\ 5 b a^{2}+3 a^{2}+a^{2} b-4 a^{2}-2 a b^{2}\).

Solution

\(\ 5 b a^{2}+3 a^{2}+a^{2} b-4 a^{2}-2 a b^{2}\)	Identify the like terms in the polynomial. Since \(\ 5 b a^{2}\) can also be written \(\ 5 a^{2} b\), it is a like term to \(\ a^{2} b\).
\(\ \begin{array}{c} (5+1) a^{2} b+(3-4) a^{2}-2 a b^{2} \\ 6 a^{2} b-a^{2}-2 a b^{2} \end{array}\)	Combine like terms using the distributive property and check to make sure all like terms have been combined.

\(\ 5 b a^{2}+3 a^{2}+a^{2} b-4 a^{2}-2 a b^{2}=6 a^{2} b-a^{2}-2 a b^{2}\)

Exercise

Simplify by collecting like terms: \(\ 4\left(x^{2} y+7 y\right)-5 y\left(3 x^{2}-y\right)-10 y\)

1. \(\ -11 x^{2} y+5 y^{2}+18 y\)
2. \(\ 4 x^{2} y+11 y-8 y x^{2}-16 y\)
3. \(\ 4 x^{2} y+18 y-15 y x^{2}+5 y^{2}\)
4. \(\ 4 x^{2} y-5 y\left(3 x^{2}-y\right)-3 y\)

Answer

1. Correct.

   \(\ \begin{array}{l}
   4\left(x^{2} y+7 y\right)-5 y\left(3 x^{2}-y\right)-10 y \text { equals } \\
   4 x^{2} y+28 y-15 y x^{2}+5 y^{2}-10 y . \text { This equals } \\
   4 x^{2} y+18 y-15 y x^{2}+5 y^{2}, \text { which equals } \\
   -11 x^{2} y+5 y^{2}+18 y
   \end{array}\)

2. Incorrect. The 4 is distributed to both terms in the parentheses by multiplying each term by 4 resulting in \(\ 28y\), not \(\ 11y\). The \(\ -5 y\) is similarly distributed to each term in the parentheses, resulting in \(\ -15 y x^{2}+5 y^{2}\). The correct answer is \(\ -11 x^{2} y+5 y^{2}+18 y\).
3. Incorrect. This polynomial can be further simplified by combining the like terms \(\ 4 x^{2} y\) and \(\ -15 y x^{2}\). The order of the variables in a term does not matter. The correct answer is \(\ -11 x^{2} y+5 y^{2}+18 y\).
4. Incorrect. Before combining like terms, you must distribute to clear the parentheses. The correct answer is \(\ -11 x^{2} y+5 y^{2}+18 y\).

Summary

Polynomials can contain more than one variable and can be evaluated in the same way as polynomials with one variable. To evaluate any polynomial, you substitute the given values for the variable and perform the computation to simplify the polynomial to a numerical value. The order of operations and integer operations must be properly applied to correctly evaluate a polynomial.

Polynomials with more than one variable can be simplified by combining like terms, as you can do with polynomials with one variable. Like terms must contain the same exact variables raised to the same exact power. In terms with more than one variable, the order in which the variables are written does not matter.