Simplifying Algebraic Expressions
Textbook Reference
| Primary source | OpenStax College Algebra 2e, Section 1.4: “Polynomials” |
| Direct link | https://openstax.org/books/college-algebra-2e/pages/1-4-polynomials |
| Supplementary | OpenStax College Algebra 2e, Section 1.1: “Real Numbers: Algebra Essentials” (distributive property) |
| Supplementary link | https://openstax.org/books/college-algebra-2e/pages/1-1-real-numbers-algebra-essentials |
Both sources are free and openly licensed. Combining like terms and the distributive property are introduced in Section 1.1 and used throughout Section 1.4.
Key idea
You can only add things that are the same kind of thing. Three apples plus two apples is five apples, but three apples plus two oranges is just three apples and two oranges.
In algebra the “kind of thing” is the variable part. The terms $3x$ and $2x$ are the same kind (“$x$ things”), so they combine into $5x$. The terms $3x$ and $2x^2$ are different kinds (“$x$ things” versus “$x$-squared things”), so they do not combine. Simplifying an expression is mostly the act of grouping the same kinds together and counting.
This means: simplifying is sorting. Sort the terms into piles by their variable part, then count each pile. The distributive property is the tool that opens parentheses so the sorting can happen.
Prerequisite Check
Before this lesson, make sure you can do all of the following:
If the distributive property is unfamiliar, review The Real Number System, where it is introduced.
Quick Reference
Key vocabulary.
| Term | Meaning | Example |
|---|---|---|
| Term | a single number or product separated by $+$ or $-$ | in $3x - 7$, the terms are $3x$ and $-7$ |
| Coefficient | the numerical factor of a term | the coefficient of $3x$ is $3$ |
| Variable part | the letters-and-exponents factor of a term | the variable part of $3x^2$ is $x^2$ |
| Like terms | terms with the same variable part | $3x^2$ and $-5x^2$ are like terms |
| Constant | a term with no variable | $7$ and $-2$ are constants |
The two moves of simplification.
- Distribute to remove parentheses: $a(b + c) = ab + ac$.
- Combine like terms by adding their coefficients: $3x + 5x = 8x$.
The deciding rule for “like”. Two terms are like terms exactly when they have the same variable raised to the same power. $4x$ and $4x^2$ are NOT like terms.
Key Concepts
1. Terms, Coefficients, and Like Terms
An expression is built from terms joined by addition or subtraction. Each term has a coefficient (the number) and a variable part (the letters and exponents). Like terms share the same variable part.
Example 1. Identify the terms and the coefficients in $4x^2 - 7x + 5$.
| Term | Coefficient | Variable part |
|---|---|---|
| $4x^2$ | $4$ | $x^2$ |
| $-7x$ | $-7$ | $x$ |
| $5$ | $5$ | none (constant) |
Note that the sign in front of a term belongs to that term: the coefficient of the middle term is $-7$, not $7$.
2. Combining Like Terms
To combine like terms, add their coefficients and keep the shared variable part. This is the distributive property read backwards: $3x + 5x = (3 + 5)x = 8x$.
Example 2. Simplify $6x + 2x - 3x$.
All three are $x$ terms. Add the coefficients: $6 + 2 - 3 = 5$. \[ 6x + 2x - 3x = 5x \]
Example 3. Simplify $4a + 3b - a + 5b$.
Sort into two piles. The $a$ pile: $4a - a = 3a$. The $b$ pile: $3b + 5b = 8b$. \[ 4a + 3b - a + 5b = 3a + 8b \] The two piles do not merge, because $a$ and $b$ are different kinds of thing.
Important: do not change the exponent when combining. $x^2 + x^2 = 2x^2$, not $x^4$. Combining like terms adds the coefficients; it never touches the variable part. The exponent stays the same.
3. The Distributive Property Removes Parentheses
The distributive property $a(b + c) = ab + ac$ lets you multiply a factor across a sum. This is how parentheses are removed so like terms can be collected.
Example 4. Simplify $3(2x + 4)$.
\[ 3(2x + 4) = 3 \cdot 2x + 3 \cdot 4 = 6x + 12 \]
Example 5. Simplify $-2(3x - 5)$.
Distribute the $-2$ to both terms, watching the signs: \[ -2(3x - 5) = (-2)(3x) + (-2)(-5) = -6x + 10 \]
Important: a negative factor flips every sign inside. When you distribute $-2$, the second product is $(-2)(-5) = +10$, not $-10$. Distributing a negative sign across a parenthesis changes the sign of each term inside.
4. Distribute, Then Combine
Most simplification problems need both moves: first distribute to clear parentheses, then combine like terms.
Example 6. Simplify $5(x + 2) - 3(x - 1)$.
Step 1 (distribute): $5(x + 2) = 5x + 10$ and $-3(x - 1) = -3x + 3$.
Step 2 (rewrite): $5x + 10 - 3x + 3$.
Step 3 (combine like terms): $x$ pile $5x - 3x = 2x$; constant pile $10 + 3 = 13$. \[ 5(x + 2) - 3(x - 1) = 2x + 13 \]
Important: the minus sign in front of a parenthesis is a $-1$. In $-3(x - 1)$, the entire group is multiplied by $-3$, so the $-1$ inside becomes $+3$. A subtraction in front of parentheses must be distributed, not dropped.
5. A Longer Worked Example
Example 7. Simplify $2x^2 + 3(x^2 - 4x) + 8x - 1$.
Step 1 (distribute): $3(x^2 - 4x) = 3x^2 - 12x$.
Step 2 (rewrite): $2x^2 + 3x^2 - 12x + 8x - 1$.
Step 3 (sort and combine):
- $x^2$ pile: $2x^2 + 3x^2 = 5x^2$.
- $x$ pile: $-12x + 8x = -4x$.
- constant pile: $-1$.
\[ 2x^2 + 3(x^2 - 4x) + 8x - 1 = 5x^2 - 4x - 1 \]
Notice the $x^2$ pile and the $x$ pile stay separate; they are different kinds of term.
Self-check: Simplify $4(2y - 3) - (y + 5)$.
Distribute: $4(2y - 3) = 8y - 12$, and $-(y + 5) = -y - 5$ (the leading minus is a $-1$). Rewrite: $8y - 12 - y - 5$. Combine: $y$ pile $8y - y = 7y$; constants $-12 - 5 = -17$. Answer: $7y - 17$.
Common Errors Summary
| Error | Example | Correction |
|---|---|---|
| Combining unlike terms | $3x + 2x^2 = 5x^2$ | Different variable parts; leave as $3x + 2x^2$ |
| Changing the exponent when combining | $x^2 + x^2 = x^4$ | Add coefficients only: $2x^2$ |
| Dropping a negative when distributing | $-2(3x - 5) = -6x - 10$ | $(-2)(-5) = +10$, so $-6x + 10$ |
| Forgetting to distribute the leading minus | $-(y + 5) = -y + 5$ | $-(y + 5) = -y - 5$ |
| Distributing to only the first term | $3(2x + 4) = 6x + 4$ | The $3$ hits both: $6x + 12$ |
Common Misconceptions
like terms can be combined by adding their exponents rather than their coefficients.
This is the multiplicative-not-additive error. When combining like terms, only the coefficients are added; the variable part remains unchanged. The expression $x^2 + x^2$ equals $2x^2$, not $x^4$. Adding exponents is the rule for multiplication, not addition: $x^2 \cdot x^2 = x^4$ because the factors are multiplied. Combining like terms is a form of counting: two copies of $x^2$ plus two more copies gives four copies, $4x^2$, just as two apples plus two apples gives four apples.
Leveled Practice
Level 1: Direct Application
Problem 1. Combine like terms: $7x + 4x - 2x$.
Show answer
All $x$ terms: $7 + 4 - 2 = 9$. Answer: $9x$.
Problem 2. Distribute: $4(3a - 2)$.
Show answer
$4 \cdot 3a - 4 \cdot 2 = 12a - 8$.
Problem 3. Simplify: $5y + 3 - 2y + 6$.
Show answer
$y$ pile: $5y - 2y = 3y$. Constants: $3 + 6 = 9$. Answer: $3y + 9$.
Level 2: Distribute and Combine
Problem 4. Simplify: $2(x + 5) + 3(x - 1)$.
Show answer
Distribute: $2x + 10 + 3x - 3$. Combine: $5x + 7$.
Problem 5. Simplify: $6 - 4(2x - 3)$.
Show answer
Distribute the $-4$: $6 - 8x + 12$. Combine constants: $-8x + 18$. (The $-4$ times $-3$ is $+12$.)
Level 3: Deeper Problems
Problem 6. Simplify: $3x^2 - 2(x^2 - 5x) - 7x$.
Show answer
Distribute the $-2$: $3x^2 - 2x^2 + 10x - 7x$. Combine: $x^2$ pile $3x^2 - 2x^2 = x^2$; $x$ pile $10x - 7x = 3x$. Answer: $x^2 + 3x$.
Problem 7. A classmate simplifies $4 - 2(x - 3)$ as $2(x - 3) = 2x - 6$, so the answer is $2x - 6$. What went wrong?
Show answer
The classmate ignored the leading $4$ and the minus sign. The factor in front is $-2$, not $+2$: $4 - 2(x - 3) = 4 - 2x + 6 = -2x + 10$. The $-2$ times $-3$ gives $+6$, and the $4$ adds to it.
Mastery Checklist
You have mastered this skill when you can do all of the following without referring to notes:
Mental Model
Think of simplifying as sorting laundry into piles and counting each pile.
Every term is an item of laundry, and its variable part is its color. You may only stack items of the same color: all the $x$ items go in one pile, all the $x^2$ items in another, all the plain numbers in a third. The distributive property is the step that empties out any bags (parentheses) so every item is loose and sortable, watching the signs as you do. Once everything is in piles, you count each pile (add the coefficients) and write the totals. Different-color piles never merge.
Connections
Within MATH 114
- Solving linear equations: before you can solve, you usually simplify each side by distributing and combining like terms.
- Polynomials: adding, subtracting, and multiplying polynomials is exactly distributing and combining like terms on a larger scale.
Toward Functions
- Evaluating and rewriting functions: putting a function rule in simplest form, or simplifying after substitution, uses these same moves.
Audience Notes
For students who find math intimidating: This is sorting and counting, two things you already do every day. The only trick is watching the minus signs when you open a parenthesis. Slow down at every minus.
For students interested in proof: “Combine like terms” is the distributive property applied in reverse, $ca + cb = c(a + b)$. Naming the property behind the move is the first step toward writing algebra you can justify line by line.
For students interested in careers: Symbolic-algebra software (the engine inside graphing calculators and computer-algebra systems) performs exactly this term-sorting automatically. Understanding the manual process makes the tool’s output readable rather than mysterious.
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