5A Simplifying Formulas and Like Terms

The key to simplifying formulas, is to understand “Terms”. A “term” is a product of numbers and or letters. A term can be a number by itself, a letter by itself, or a product of letters and numbers. Here are some examples of terms:

12b

-11xy

W

-4

$$7x^2$$

As you can see, a term has two parts: a numerical coefficient (number part) and most of the time a variable part (letter). Let’s see if we can separate these terms into their numerical coefficients and variable part.

12b: We see that 12 is the numerical coefficient and b is the variable (letter) part

-11xy: We see that -11 is the numerical coefficient and xy is the variable (letter) part

W: This is an interesting term as we don’t see a number part. There is a number part though. Since W = 1W, We see that 1 is the numerical coefficient and of course W is the variable (letter) part.

-4: This is also an interesting case as there is no variable part. This is a special term called a constant or constant term. Constants have a number part (-4) but no variable part.

$7x^2$: We see that 7 is the numerical coefficient and $x^2$ is the variable (letter) part.

The degree of a term is the exponent on the variable part. So since W = $W^1$, W is a first degree term. Since $7x^2$ has a square on the variable, this is a second degree term. Notice the number does not influence the degree of a term. A constant like -4 has no variable so it is considered degree zero. Products of letters are tricky. We add the degree of all the letters. So since $3a^{2}bc=3a^2{b}^1{c}^1$, the degree is 2+1+1 = 4. It is a 4^th degree term.

Try the following examples with your instructor. For each term, identify the numerical coefficient and the variable part (if it has one). Also give the degree of the term.

One of the key things to know about terms is that we can combine terms into a single term by addition or subtraction only when they have the same variable part. So we can combine x terms with x terms and $r^2$ with $r^2$ and so on. Terms with the same variable part are called “like terms.” To add or subtract like terms we add or subtract the numerical coefficients and keep the variables (letters) the same.

Look at the example of 5a+3a. Are these like terms? They both have the exact same letter part, so they are like terms. Think of it like 5 apples plus 3 apples. We would have 8 apples, not 8 apples squared or 8 double apples. So 5a + 3a = 8a. We can combine the like terms and keep the letter part the same.

Look at the example of 7a + 2b. Are these like terms? Since they have different letter parts, they are not like terms. Hence we cannot combine them into a single term. Think of it like 7 apples plus 2 bananas. That will not equal 9 apple/bananas. It is just 7 apples and 2 bananas. That is a good way of thinking about adding or subtracting terms that don’t have the same letter part. Hence 7a + 2b = 7a + 2b. They stay separate. In fact, many formulas have two or more terms that cannot be combined. 7a + 2b is as simplified as we can make it.

We even have special names for formulas that tell us how many terms it has. A formula with only 1 term is called a “monomial”. A formula with exactly 2 terms is called a “binomial”.
A formula with exactly 3 terms is called a “trinomial”.

Try and simplify the following formulas with your instructor. After it is simplified, count how many terms the simplified form has. Then name the formula as a monomial, binomial or trinomial.

Practice Problems Section 5A

For each term, identify the numerical coefficient and the variable part (if it has one). Also give the degree of the term.

1. $$9L$$

   Check answer

   Numerical Coefficient = 9; Variable = L; 1^st Degree

2. $$-3r^4$$

3. $$18$$

   Check answer

   Numerical Coefficient = 18; No Variable part; Degree Zero (Constant)

4. $$y^3$$

5. $$-r^2$$

   Check answer

   Numerical Coefficient = $-1$; Variable = $r^2$; 2^nd Degree

6. $$-12p$$

7. $$23v^5$$

   Check answer

   Numerical Coefficient = $23$; Variable = $v^5$; 5^th Degree

8. $$x^7$$

9. $$5^2$$

   Check answer

   Numerical Coefficient = 25; No Variable part; Degree Zero (Constant)

10. $$-w^2$$

11. $$-19h^2$$

    Check answer

    Numerical Coefficient = $-19$; Variable = $h^2$; 2^nd Degree

12. $$3\text{abc}$$

13. $$-3x^2{y}^2$$

    Check answer

    Numerical Coefficient = $-3$; Variable = $x^2{y}^2$; 4^th Degree

14. $$m^{2}n$$

15. $$-v{w}^2$$

    Check answer

    Numerical Coefficient = $-1$; Variable = $v{w}^2$; 3^rd Degree

16. $$-7b^3$$

17. $$y^6$$

    Check answer

    Numerical Coefficient = $1$; Variable = $y^6$; 6^th Degree

18. $$-17$$

19. $$-p^8$$

    Check answer

    Numerical Coefficient = $-1$; Variable = $p^8$; 8^th Degree

20. $$19k^5$$

21. $$-13wx{y}^2$$

    Check answer

    Numerical Coefficient = $-13$; Variable = $wx{y}^2$; 4^th Degree

22. $$19a^3{b}^2$$

23. $$p^4{q}^2$$

    Check answer

    Numerical Coefficient = $1$; Variable = $p^4{q}^2$; 6^th Degree

24. $$-v^2{w}^2$$

Simplify the following formulas by adding or subtracting the like terms if possible. Count how many terms the simplified form has and then name the formula as a monomial, binomial or trinomial.

25. $$-3a+11a$$

    Check answer

    $8a$; 1 term; Monomial

26. $$12m+7m$$

27. $$-6v-+14v$$

    Check answer

    $-20v$; 1 term; Monomial

28. $$5x-14x$$

29. $$17m+9m-8m$$

    Check answer

    $18m$; 1 term; Monomial

30. $$-13p+-9p+5p$$

31. $$3y-8$$

    Check answer

    $3y-8$; 2 terms; Binomial

32. $$6a+4b+9b$$

33. $$-2p+-8p-3m$$

    Check answer

    $-10p-3m$; 2 terms; Binomial

34. $$8v-17v-12v$$

35. $$4a+6b-8c$$

    Check answer

    $4a+6b-8c$; 3 terms; Trinomial

36. $$3g-7h+10g-5h$$

37. $$5w-8x+3y$$

    Check answer

    $5w-8x+3y$; 3 terms; Trinomial

38. $$x^2+9x^2-7x+1$$

39. $$-2w^2+4w-8w$$

    Check answer

    $-2w^2-4w$; 2 terms; Binomial

40. $$x^3+7x-9$$

41. $$4m^3+9m^2-10m^3-7m^2$$

    Check answer

    $-6m^3+2m^2$; 2 terms; Binomial

42. $$5y-7y^2+8y-3$$

Adapted from Preparing for Algebra and Statistics, Third Edition, by Matt Teachout. License: CC BY 4.0. Copyright and sources.