### Learning Objectives

- Simplify polynomials by collecting like terms

## 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.