Combine Like Terms: Merge Coefficients of Identical Powers

This guide sits inside the Algebra study map, where you can see the neighboring moves, models, and next-step guides that connect this principle to the rest of algebra.

On this page: The Principle | Conditions | Failure Modes | EE Questions | Retrieval Practice | Practice Ground | Solve a Problem | FAQ

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The Principle

The move: Add or subtract the coefficients of terms whose variable-and-exponent factors are identical, replacing two or more terms with a single term.

The invariant: This preserves the value of the expression for every allowed assignment of the variables: combining like terms produces an equivalent expression.

Pattern: $ax + bx = (a + b)x$

Legal ✓	Illegal ✗
$3x^2 + 5x^2 = 8x^2$	$3x + 5x^2 \not\to 8x^2$ — exponents differ; cannot merge

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Conditions of Applicability

Condition: Same variables and exponents

Before applying, check: Verify that the terms share the same full variable-and-exponent factor, not just the same variable letter.

• $3x$ and $5x^2$ cannot be combined: the exponents differ ($1 \neq 2$).
• $4x$ and $3y$ cannot be combined: the variable names differ.
• $2xy$ and $5x^2y$ cannot be combined: the exponent on $x$ differs.

Want the complete framework behind this guide? Read Masterful Learning.

When an expression needs rearrangement before terms can be merged, compare this move with Distributive Property. After the terms are collected cleanly, Factor Common Term is often the next algebra move that reveals more structure.

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Common Failure Modes

Failure mode: combining terms with different exponents — e.g., writing $3x + 5x^2$ as $8x^2$ → the linear $3x$ term is silently destroyed, producing a structurally wrong expression.

Debug: match the FULL variable-and-exponent factor exactly before adding coefficients; $x$ and $x^2$ are different factors.

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Elaborative Encoding

Use these questions to build deep understanding. (See Elaborative Encoding for the full method.)

Within the Principle

• What does “coefficient” mean in $3x^2 + 5x^2$, and why do you add the coefficients $3$ and $5$ while leaving $x^2$ unchanged?
• Why is $3x + 5x = 8x$, but $3x + 5y$ cannot be reduced to a single term?

For the Principle

• How do you decide which terms in a multi-variable expression such as $2x^2y + 3xy^2 + 4x^2y$ are “like”?
• What would go wrong if you applied this move to $3x + 5x^2$?

Between Principles

• How does the Distributive Property justify why Combine Like Terms works: which algebraic identity makes $ax + bx = (a+b)x$ valid?

Generate an Example

• Write an expression with four terms, exactly two of which are like terms, that requires this principle to simplify — and identify which pair qualifies.

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Retrieval Practice

Answer from memory, then click to reveal and check. (See Retrieval Practice for the full method.)

State the move in one sentence: _____Add or subtract the coefficients of terms that have the same variables raised to the same exponents.

Write the canonical pattern: _____$ax + bx = (a + b)x$

State the canonical condition: _____Same variables and exponents

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Practice Ground

Use these exercises to build move-selection fluency. (See Self-Explanation for how to use worked examples effectively.)

Procedure Walkthrough

Starting from $4x^2 - 7x + 3x^2 + 2x - 1$, collect all like terms to reach simplified form.

Step	Expression	Operation
0	$4x^2 - 7x + 3x^2 + 2x - 1$	—
1	$(4x^2 + 3x^2) + (-7x + 2x) - 1$	Group the $x^2$ terms together and the $x$ terms together — $x^2$ and $x$ are different factors, so they must stay separate
2	$7x^2 + (-5x) - 1$	Combine: $4+3=7$ for $x^2$ terms; $-7+2=-5$ for $x$ terms
3	$7x^2 - 5x - 1$	Rewrite $+(-5x)$ as $-5x$

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Drills

Format A — Forward step

Apply the principle once.

$5x + 3x$

Reveal

Add the coefficients ($5 + 3 = 8$); $x$ stays unchanged:

$8x$

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Apply the principle once.

$9y^2 - 4y^2$

Reveal

Subtract the coefficients ($9 - 4 = 5$); $y^2$ stays unchanged:

$5y^2$

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Apply the principle once.

$7ab - 3ab$

Reveal

Subtract the coefficients ($7 - 3 = 4$); $ab$ stays unchanged:

$4ab$

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Apply the principle once.

$2x^3 + 4x^3 + x^3$

Reveal

Add all three coefficients ($2 + 4 + 1 = 7$); $x^3$ stays unchanged:

$7x^3$

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Apply the principle once.

$6m - 2m + 5m$

Reveal

Combine left to right: $6 - 2 + 5 = 9$:

$9m$

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Identify which terms are like, then apply the principle.

$3x + 5x^2 + 2x$

Reveal

$3x$ and $2x$ are like terms (both have $x^1$). $5x^2$ is NOT like them (exponent differs).

Combine only the $x$ terms:

$5x^2 + 5x$

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Reject the invalid rewrite. What is wrong?

$2x + 3x^2 \not\to 5x^2$

Reveal

$2x$ and $3x^2$ are not like terms. They do not share the same full variable-and-exponent factor: one carries $x$, the other carries $x^2$. Adding their coefficients destroys the $2x$ term entirely.

No combination is valid here. A collected form simply keeps both terms: $3x^2 + 2x$.

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Format E — Canonicalization

Rewrite in simplified (collected) form.

$3x + 5y + 2x - y$

Reveal

Group by variable: $(3x + 2x) + (5y - y) = 5x + 4y$

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Rewrite in simplified form.

$4a^2 + a - 3a^2 + 2a$

Reveal

$a^2$ terms: $4 - 3 = 1$. $a$ terms: $1 + 2 = 3$:

$a^2 + 3a$

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Rewrite in simplified form.

$2p^3 - p + 4p^3 - 3p$

Reveal

$p^3$ terms: $2 + 4 = 6$. $p$ terms: $-1 - 3 = -4$:

$6p^3 - 4p$

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Rewrite in simplified form.

$7t^2 - 2t + 3t - 5t^2$

Reveal

$t^2$ terms: $7 - 5 = 2$. $t$ terms: $-2 + 3 = 1$:

$2t^2 + t$

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Rewrite in simplified form.

$x + 2x^2 - 3x + x^2$

Reveal

$x^2$ terms: $2 + 1 = 3$. $x$ terms: $1 - 3 = -2$:

$3x^2 - 2x$

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Solve a Problem

Apply what you’ve learned with Problem Solving.

Problem: Starting from $5x^2 - 3x + 2x^2 + 7x - 4$, collect all like terms to reach simplified form.

Full solution

Step	Expression	Move
0	$5x^2 - 3x + 2x^2 + 7x - 4$	—
1	$(5x^2 + 2x^2) - 3x + 7x - 4$	Group the $x^2$ terms
2	$7x^2 - 3x + 7x - 4$	Combine $x^2$ terms: $5 + 2 = 7$
3	$7x^2 + (-3x + 7x) - 4$	Group the $x$ terms
4	$7x^2 + 4x - 4$	Combine $x$ terms: $-3 + 7 = 4$

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FAQ

What is Combine Like Terms?

Combine Like Terms is the algebraic rule that lets you replace two or more terms with a single term by adding or subtracting their coefficients, provided the terms share identical variable-and-exponent factors. For example, $3x + 5x = 8x$ because both terms carry the same factor $x$.

When is Combine Like Terms valid?

The move is valid whenever two or more terms share exactly the same variables raised to exactly the same exponents. $3x^2$ and $7x^2$ qualify (both have $x^2$); $3x$ and $7x^2$ do not (exponents $1$ and $2$ differ).

What goes wrong if I ignore the condition?

Combining terms with different exponents destroys one term. Writing $3x + 5x^2 = 8x^2$ discards the $3x$ contribution, producing a structurally incorrect expression whose value differs from the original for almost every value of $x$.

How does Combine Like Terms relate to the Distributive Property?

The Distributive Property ($a(b+c) = ab + ac$) is the algebraic justification: $ax + bx = (a+b)x$ by factoring out $x$. Combine Like Terms is the move you execute; the Distributive Property is the identity that makes it valid.

Does Combine Like Terms extend to expressions with several variables?

Yes — the condition extends naturally. $3xy + 5xy = 8xy$ (both share the factor $xy$), but $3xy$ and $5x^2y$ cannot be merged because the exponent on $x$ differs. Treat the full variable-and-exponent product as the unit of comparison.

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How This Fits in Unisium

Combine Like Terms appears at the start of almost every multi-step algebra problem. Unisium builds fluency with this move through repeated forward-step and canonicalization drills — the same format used in the Practice Ground above. The goal is automatic recognition: seeing $4x^2 + 3x^2$ and immediately writing $7x^2$ without pausing to think. The failure-mode drill (spotting when terms are NOT like) trains the safeguard that prevents coefficient errors from propagating through longer chains.

Explore further:

• Elaborative Encoding — Build deep understanding of why the variable-and-exponent condition matters
• Retrieval Practice — Make the pattern and condition instantly accessible
• Self-Explanation — Use the Practice Ground drills at maximum effectiveness

Ready to master Combine Like Terms? Start practicing with Unisium or explore the full learning framework in Masterful Learning.

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