Eighth Grade Mathematics: Distributive Property and Combining Terms
Algebraic expressions often appear in complicated, expanded forms with multiple sets of parentheses and scattered variable terms that obscure the underlying relationships. Before you can solve complex multi-step equations or analyze functions, you must simplify these expressions into their cleanest, most compact equivalents. In this chapter, you will master expanding expressions using the distributive property, recognizing like terms through their variable bases and powers, and combining terms systematically while navigating tricky sign rules.
The Mechanics of the Distributive Property
The distributive property asserts that multiplying a single factor outside parentheses by an expression inside parentheses requires multiplying that outer factor across every individual term inside the grouping.
The Distributive Property of Multiplication:
a * (b + c) = a * b + a * c
a * (b - c) = a * b - a * c
Geometrically, this can be visualized as finding the total area of a large rectangle partitioned into two smaller adjacent rectangles sharing the exact same height a.
Area Representation of a * (b + c):
b c
+-------------+---------------+
| | |
a | Area = ab | Area = ac |
| | |
+-------------+---------------+
Total Area = ab + ac = a * (b + c)
Distributing Negative Factors
The single most common algebraic error occurs when distributing a negative coefficient across parentheses. The negative sign belongs to the factor and must be multiplied into every single term, reversing all internal signs.
Example 1: Simplify -4 * (2x - 5)
Multiply -4 by (2x): -4 * (2x) = -8x
Multiply -4 by (-5): -4 * (-5) = +20
Result: -8x + 20
Example 2: Simplify -(3x - 7)
A leading negative implies a factor of -1:
-1 * (3x - 7) = -1*(3x) + (-1)*(-7) = -3x + 7
Taking time to write out the intermediate sign products prevents careless subtraction and negative sign errors.
Identifying and Combining Like Terms
Like terms are defined as algebraic terms that have the exact same variable bases raised to the exact same exponential powers. Constant numerical terms without variables are also like terms with each other.
Term Classification:
Terms Like or Unlike? Reason
4x and -9x Like Terms Identical variable x with exponent 1
5x^2 and 3x^2 Like Terms Identical variable x with exponent 2
7x and 7y Unlike Terms Different variables (x vs y)
2x^2 and 6x Unlike Terms Different exponents (2 vs 1)
-11 and 18 Like Terms Both are pure constants
The Algebraic Rule for Combining Like Terms
To combine like terms, you apply the distributive property in reverse: factor out the common variable expression and perform arithmetic addition or subtraction strictly on their numerical coefficients.
Combining Process:
8x + 5x = (8 + 5)*x = 13x
4x^2 - 9x^2 = (4 - 9)*x^2 = -5x^2
Notice that the exponent does not change when combining like terms. A frequent misconception is writing 3x + 4x = 7x^2. Adding like terms counts how many groups of the variable you possess; it does not multiply the variables together.
Simplifying Multi-Step Algebraic Expressions
To simplify complex expressions containing multiple operations, follow a systematic order of execution: first expand all parentheses using the distributive property, then rearrange and group like terms using commutative and associative laws, and finally combine the coefficients.
Simplify: 3*(4x - 2) - 5*(2x - 6) + 4
Step 1: Distribute the 3 across the first set of parentheses:
3*(4x) - 3*(2) = 12x - 6
Step 2: Distribute the -5 across the second set of parentheses:
-5*(2x) - 5*(-6) = -10x + 30
Step 3: Assemble the expanded expression:
12x - 6 - 10x + 30 + 4
Step 4: Group like variable terms and constant terms:
(12x - 10x) + (-6 + 30 + 4)
Step 5: Combine coefficients:
2x + 28
Chapter Exercises
Exercise 1: Fully simplify the algebraic expression -2(5x - 8) + 4(3x - 1). Show both distributive multiplications and the grouping of like terms in paragraph form.
Exercise 2: Simplify the compound expression 6 - 3*(2x - 4) + 5x - (x - 9). Pay careful attention to the order of operations and the subtraction of grouped terms.
Exercise 3: A student simplifies the expression 5(2x + 3) - 2(x - 4) and arrives at the answer 8x - 1. Identify the specific errors the student committed, explaining why they occurred, and provide the correct simplified expression.
Solutions and Explanations
Solution 1: To simplify -2(5x - 8) + 4(3x - 1), begin by applying the distributive property to both terms. Distribute the -2 into (5x - 8): multiplying -2 by 5x yields -10x, and multiplying -2 by -8 yields +16, producing -10x + 16. Next, distribute the +4 into (3x - 1): multiplying 4 by 3x yields +12x, and multiplying 4 by -1 yields -4, producing +12x - 4. Combining these expanded pieces gives -10x + 16 + 12x - 4. Now, group like terms together: combine the x-terms (-10x + 12x) and the constant terms (16 - 4). Adding the coefficients of the variable terms gives (-10 + 12)x = 2x. Adding the constants gives 16 - 4 = 12. The final, fully simplified expression is 2x + 12.
Solution 2: To simplify 6 - 3*(2x - 4) + 5x - (x - 9), first follow the order of operations: do not subtract 3 from 6 first, because multiplication takes precedence. Distribute the -3 into (2x - 4): -3 * (2x) = -6x, and -3 * (-4) = +12. Next, treat the expression -(x - 9) as distributing a factor of -1: -1 * (x) = -x, and -1 * (-9) = +9. Rewriting the entire expression with all parentheses removed yields 6 - 6x + 12 + 5x - x + 9. Now group the like variable terms: -6x + 5x - x. Combining their coefficients yields (-6 + 5 - 1)x = -2x. Next, group the constant terms: 6 + 12 + 9 = 27. Putting the terms together yields the simplified expression -2x + 27.
Solution 3: The student made two distinct distribution errors, both involving negative signs. First, when distributing the 5 across (2x + 3), the student correctly obtained 10x + 15. However, when distributing the -2 across (x - 4), the student multiplied -2 by x to get -2x, but incorrectly calculated -2 * (-4) as -8 instead of +8 (forgetting that multiplying two negative numbers produces a positive number). Second, the student appears to have subtracted 15 - 8 = 7, but then made an additional subtraction error or combined constants incorrectly to get -1. The correct distribution gives 10x + 15 - 2x + 8. Combining the like variable terms 10x - 2x yields 8x. Combining the constant terms 15 + 8 yields +23. Therefore, the correct simplified expression is 8x + 23, not 8x - 1.