Simplify Polynomials

Learning Objectives

  • Simplify polynomials by collecting like terms
Apple sitting next to an Orange

Apple and Orange

Simplify polynomials by collecting like terms

A polynomial may need to be simplified. One way to simplify a polynomial is to combine the like terms if there are any. Two or more terms in a polynomial are like terms if they have the same variable (or variables) with the same exponent. For example, [latex]3x^{2}[/latex] and [latex]-5x^{2}[/latex] are like terms: They both have x as the variable, and the exponent is 2 for each. However, [latex]3x^{2}[/latex] and [latex]3x[/latex] are not like terms, because their exponents are different.

Here are some examples of terms that are alike and some that are unlike.

Term Like Terms UNLike Terms
[latex]a[/latex] [latex]3a, \,\,\,-2a,\,\,\, \frac{1}{2}a[/latex] [latex]a^2,\,\,\,\frac{1}{a},\,\,\, \sqrt{a}[/latex]
[latex]a^2[/latex] [latex]-5a^2,\,\,\,\frac{1}{4}a^2,\,\,\, 0.56a^2[/latex] [latex]\frac{1}{a^2},\,\,\,\sqrt{a^2},\,\,\, a^3[/latex]
[latex]ab[/latex] [latex]7ab,\,\,\,0.23ab,\,\,\,\frac{2}{3}ab,\,\,\,-ab[/latex] [latex]a^2b,\,\,\,\frac{1}{ab},\,\,\,\sqrt{ab}[/latex]
[latex]ab^2[/latex]  [latex]4ab^2,\,\,\, \frac{ab^2}{7},\,\,\,0.4ab^2,\,\,\, -a^2b[/latex]  [latex]a^2b,\,\,\, ab,\,\,\,\sqrt{ab^2},\,\,\,\frac{1}{ab^2}[/latex]

Example

Which of these terms are like terms?

[latex]7x^{3}7x7y-8x^{3}9y-3x^{2}8y^{2}[/latex]

You can use the distributive property to simplify the sum of like terms. Recall that the distributive property of addition states that the product of a number and a sum (or difference) is equal to the sum (or difference) of the products.

[latex]2\left(3+6\right)=2\left(3\right)+2\left(6\right)[/latex]

Both expressions equal 18. So you can write the expression in whichever form is the most useful.

Let’s see how we can use this property to combine like terms.

Example

Simplify [latex]3x^{2}-5x^{2}[/latex].

You may have noticed that combining like terms involves combining the coefficients to find the new coefficient of the like term. You can use this as a shortcut.

Example

Simplify [latex]6a^{4}+4a^{4}[/latex].

When you have a polynomial with more terms, you have to be careful that you combine only like terms. If two terms are not like terms, you can’t combine them.

Example

Simplify [latex]3x^{2}+3x+x+1+5x[/latex]

Summary

Polynomials are algebraic expressions that contain any number of terms combined by using addition or subtraction. A term is a number, a variable, or a product of a number and one or more variables with exponents. Like terms (same variable or variables raised to the same power) can be combined to simplify a polynomial. The polynomials can be evaluated by substituting a given value of the variable into each instance of the variable, then using order of operations to complete the calculations.