1. Introduction
A common practice in mathematics is to write answers in simplest form. However, not every expression can be combined or simplified. For example, consider the expression “5 apples and 9 cars.” This cannot be simplified to “14 apples/cars,” because apples and cars are not alike and thus cannot be combined. This means that the expression “5 apples and 9 cars” is already in its simplest form. In contrast, the expression “5 apples and 8 more apples” can be combined and simplified to “13 apples” since 5 apples + 8 apples = 13 apples. Likewise, polynomial terms can be combined if they are what we call “like terms.”
1.2 Skill 5: Combining Like Terms in Polynomials

2. Key Concepts
A term is a number, a variable, or the product of a number and variable(s). For example, the expression 8x2 + 9x – 6 has three terms. The first is 8x2, the second is 9x, and the third is –6. Notice that the negative sign is part of the last term.
A monomial is an expression with one term, consisting of a number, a variable, or the product of a number and variable(s). For example, 16y is a monomial because it is only one term.
A polynomial is a monomial or the sum of monomials. It is a collection of terms that are added together. For example, 9y3 – 12y + 3 is a polynomial.
1.2 Skill 5: Combining Like Terms in Polynomials

3. Key Concepts, continued
Like terms are terms that have the same variable(s) raised to the same power. For example, 8m4 and –5m4 are like terms. Numeric quantities (numbers without variables) are always like terms. For example, 9 and 2 are like terms.
Only like terms can be added or subtracted. For example, 8x + 10x can be added, but 9x2 + 4x cannot. Notice that 8x and 10x are raised to the same power, but 9x2 and 4x are not.
1.2 Skill 5: Combining Like Terms in Polynomials

4. Key Concepts, continued
To combine like terms, add or subtract the coefficients (the numbers being multiplied by the variables) and keep the variable(s) and exponents the same. For example, 7a3 + 5a3 = 12a3.
Terms with no coefficient have an understood coefficient of 1. For example, x is the same as 1x.
Note that it is common to write polynomials in alphabetical order. Follow this rule when multiple variables in a polynomial have the same power.
1.2 Skill 5: Combining Like Terms in Polynomials

5. Key Concepts, continued
A polynomial is considered to be in standard form when the terms are written with the exponents in descending order and the constant term last. For example, the polynomial 6x2 + 2x3 + 5x + 3 is not in standard form because the term with highest exponent is not listed first. Rearrange the terms so the exponents are in descending order: 2x3 + 6x2 + 5x + 3. Now the polynomial is in standard form.
1.2 Skill 5: Combining Like Terms in Polynomials

6. Guided Practice Example 3
Vlad owns a paving business. He quotes prices to his customers based on area. He has determined that the area of a new customer’s driveway is equal to 9x2 – 9x – 15x + 15 square feet. Simplify this expression by combining like terms.
1.2 Skill 5: Combining Like Terms in Polynomials

7. Guided Practice: Example 3, continued
Determine which terms are like terms.
Like terms are terms that contain the same variables raised to the same power.
The expression 9x2 – 9x – 15x + 15 contains one set of like terms.
The terms –9x and –15x are like terms because they have the same variable (x) raised to the same power (1).
1.2 Skill 5: Combining Like Terms in Polynomials

8. Guided Practice: Example 3, continued Combine each set of like terms.
When combining like terms, add or subtract the coefficients (the numbers being multiplied by the variables) and keep the variables and exponents the same.
To combine –9x and –15x, first add –9 and –15:
–9 + (–15) = –23
Keep the variable and exponent (x) the same:
–9x + (–15x) = –23x
Combining the like terms gives us –23x.
1.2 Skill 5: Combining Like Terms in Polynomials

9. Guided Practice: Example 3, continued
Rewrite the expression with the combined like terms.
In the expression 9x2 – 9x – 15x + 15, the like terms are already grouped together, so there’s no need to rearrange them. Rewrite the expression by substituting the combined terms:
9x2 – 9x – 15x Original polynomial
9x2 (– 23x) Substitute –23x for –9x – 15x.
Since 9x2 and 15 did not have any like terms, they remain the same.
1.2 Skill 5: Combining Like Terms in Polynomials

10. Guided Practice: Example 3, continued
The expression 9x2 – 9x – 15x + 15 can be simplified to 9x2 – 23x + 15.
1.2 Skill 5: Combining Like Terms in Polynomials

11. Guided Practice: Example 3, continued
1.2 Skill 5: Combining Like Terms in Polynomials