Are 3x and 3x^2 like terms?

No. Like terms must match in both the variable and the exponent. 3x counts groups of x, while 3x^2 counts groups of x · x, so they cannot be added into one term. You can verify this: at x = 4, 3x = 12 but 3x^2 = 48. The expression 3x + 3x^2 is already simplified.

Why do all the signs flip when I subtract a parenthesis?

Because a bare minus sign in front of parentheses is really a multiplier of -1, and that -1 distributes to every term inside. So -(2x - 9) becomes (-1)(2x) + (-1)(-9) = -2x + 9. Writing the -1 in explicitly before you distribute prevents the most common error in this lesson.

Do I always have to distribute first?

Distribute before combining anything that sits outside the parentheses with something inside — that part is required by the order of operations. But if the terms inside the parentheses are themselves like terms, you may simplify inside first. In 3(2x + 5x), you can combine to get 3(7x) = 21x, or distribute to get 6x + 15x = 21x. Both routes give the same result.

How do I know my simplified expression is correct?

Substitute a convenient number for the variable — x = 2 or x = 3 works well, but avoid x = 0 and x = 1 since they can hide sign errors — and evaluate both the original expression and your answer. Equivalent expressions must give the same value. If the two values differ, retrace your distribution step first; a dropped negative is the usual cause.

ALG1-1.3

Simplifying Expressions: Distributive Property & Like Terms

Learn to simplify algebra expressions with the distributive property (including negative multipliers) and combining like terms, with worked steps and common error fixes.

What you'll do in this lesson

A voice-first session with the Crimsora tutor on Simplifying Expressions: Distributive Property & Like Terms, then targeted practice and FRQs — with the tutor adapting to where you get stuck.

Start Simplifying Expressions: Distributive Property & Like Terms free

What this lesson covers

You have already evaluated expressions and translated words into algebra. Now you will rewrite expressions so they are shorter and easier to work with — without changing their value. That is what "simplify" means:

3(x+4)-2x

and

x+12

look different, but plug in any number for

x

and they produce the same output every single time.

Two tools do almost all of the work: the distributive property, which clears parentheses, and combining like terms, which collects the pieces that match. The hard part is not the idea — it is the bookkeeping, especially when a negative sits in front of a parenthesis. This guide walks through the structure of a term, the mechanics of distributing, the rule for what counts as "like," and the specific places where careful students still slip up.

Terms, Coefficients, and What "Simplify" Really Means

An algebraic expression is built from terms — chunks separated by plus or minus signs. In

7x^2 - 4x + 9

, the terms are

7x^2

-4x

, and

9

. Notice that the sign in front belongs to the term. Reading

-4x

as a negative term (instead of "minus four x") saves enormous trouble later.

Each term has a coefficient (the number multiplying the variable part) and a variable part (the letters with their exponents). In

-4x

, the coefficient is

-4

and the variable part is

x

. A term like

x

has an invisible coefficient of

1

;

-x

has coefficient

-1

. A term with no variable at all, like

9

, is a constant.

To simplify means to write an equivalent expression with as few terms as possible and no unnecessary parentheses. Equivalent is the key word: the two expressions must give identical values for every replacement of the variable. Simplifying is not solving — there is no equals sign and no answer like

x = 5

. Your result is another expression.

Expression	Number of terms	Coefficients
$5x + 3$	2	$5$ , and the constant $3$
$-x^2 + 7x - 2$	3	$-1$ , $7$ , and the constant $-2$
$6(x - 4)$	1 (a product)	$6$ times a two-term factor

That last row matters.

6(x-4)

is a single term until you distribute, because multiplication holds it together. Recognizing structure — is this a sum of terms, or a product? — tells you which tool to reach for first.

The Distributive Property, Including Negative Multipliers

The distributive property says

a(b + c) = ab + ac

. The outside factor multiplies every term inside, not just the first one. It also works with subtraction:

a(b - c) = ab - ac

, since subtracting is adding the opposite.

4(3x + 5) = 4 \cdot 3x + 4 \cdot 5 = 12x + 20

Negative multipliers are where most errors live. When the factor outside is negative, every sign inside flips:

-2(5x - 7) = (-2)(5x) + (-2)(-7) = -10x + 14

Students very often write

-10x - 14

here, forgetting that negative times negative is positive. Slow down and multiply the signs deliberately, term by term.

A subtler case is a bare minus sign in front of parentheses, as in

9 - (4x - 3)

. There is no visible number, but the minus sign means

-1

is the multiplier:

9 - 1(4x - 3) = 9 - 4x + 3 = 12 - 4x

Writing the invisible

-1

explicitly, every time, is the single best habit for this lesson.

Setup	Distributed	Frequent wrong answer
$3(x + 6)$	$3x + 18$	$3x + 6$ (forgot second term)
$-5(x + 2)$	$-5x - 10$	$-5x + 10$
$-4(x - 3)$	$-4x + 12$	$-4x - 12$
$-(2x - 9)$	$-2x + 9$	$-2x - 9$

The property runs both directions.

12x + 20

can be rewritten as

4(3x + 5)

by pulling out the common factor

4

. That reverse move is factoring, and you will lean on it heavily in later units.

Combining Like Terms

Like terms have exactly the same variable part — same letters, same exponents. Only their coefficients may differ.

3x

and

-8x

are like terms.

3x

and

3x^2

are not, because

x

and

x^2

are different variable parts.

5xy

and

2yx

are like terms, since multiplication can be reordered and both have variable part

xy

.

To combine, add the coefficients and keep the variable part unchanged:

3x - 8x = (3 - 8)x = -5x

Why does this work? The distributive property again, read backwards:

3x - 8x = (3-8)x

. Combining like terms is not a separate rule; it is factoring out the shared variable.

A classic misconception is changing the exponent:

x^2 + x^2

2x^2

, not

x^4

. You are counting how many

x^2

blocks you have, not multiplying them. Similarly,

4x + 3

cannot be combined into

7x

— three ones and four x's are different kinds of objects, and if you test

x = 2

you get

11

versus

14

, which proves they are not equivalent.

A reliable procedure: after distributing, underline or circle each family of like terms with a different mark, dragging the sign in front along with each term. Then add each family separately.

6x - 4 + 2x + 9 \;\to\; (6x + 2x) + (-4 + 9) \;\to\; 8x + 5

By convention, write the result in descending order of exponent with the constant last:

8x + 5

, not

5 + 8x

. Both are correct, but the standard order makes later work easier to read.

Multi-Step Simplification and Where Answers Go Wrong

Most problems in this lesson mix both tools. The order is nearly always the same: clear the innermost parentheses by distributing, then combine like terms, then write the terms in standard order.

Consider

8 - 3(2x - 5) + 7x

. A very common wrong first move is subtracting

3

from

8

to get

5(2x-5)

. That violates the order of operations — the

3

is attached to the parentheses by multiplication, so it must distribute before anything else touches it. Distribute first:

8 - 6x + 15 + 7x

, then combine:

x + 23

.

Nested parentheses work from the inside out. In

2[3x + 4(x - 1)]

, handle

4(x-1) = 4x - 4

first, giving

2[3x + 4x - 4] = 2[7x - 4] = 14x - 8

Step	What you do	Watch for
1	Distribute every factor over its parentheses	Negative multipliers; invisible $-1$
2	Rewrite as a plain sum of terms	Keep each sign attached to its term
3	Group like terms	$x$ and $x^2$ are not alike
4	Add coefficients	Variable part never changes
5	Write in descending order	Constant goes last

To check your work, pick a value like

x = 3

, evaluate the original expression and your simplified one, and compare. If they disagree, an error is hiding somewhere — usually a dropped negative. This substitution check catches almost every mistake in under a minute and is worth doing on any problem you are unsure about.

Key terms

Term.: A single number, variable, or product of numbers and variables, separated from other terms by plus or minus signs. The sign in front belongs to the term.
Coefficient.: The numerical factor of a term. In $-7y$ the coefficient is $-7$ ; in $x$ it is $1$ .
Constant.: A term with no variable, such as $9$ or $-4$ . Its value never changes.
Like terms.: Terms with identical variable parts, including identical exponents, such as $5x^2$ and $-2x^2$ . Only like terms may be combined.
Distributive property.: The rule $a(b+c) = ab + ac$ , which lets you multiply an outside factor by every term inside parentheses.
Equivalent expressions.: Two expressions that produce the same value for every allowed replacement of the variable, such as $3(x+2)$ and $3x+6$ .
Simplify.: To rewrite an expression in an equivalent form with no unnecessary parentheses and the fewest possible terms.
Factoring out.: Reversing distribution by pulling a common factor to the outside, as in $12x + 20 = 4(3x+5)$ .

Worked example

Simplify completely:

5(2x - 3) - 2(4x - 7) + 3x

Step 1 — Distribute the first factor. Multiply

5

by each term inside:

5 \cdot 2x = 10x

and

5 \cdot (-3) = -15

. So

5(2x-3) = 10x - 15

.

Step 2 — Distribute the second factor, sign included. The multiplier here is

-2

, not

2

. Multiply:

(-2)(4x) = -8x

and

(-2)(-7) = +14

. So

-2(4x-7) = -8x + 14

. The most common slip is writing

-14

; negative times negative is positive.

Step 3 — Rewrite the whole expression as a sum of terms.

10x - 15 - 8x + 14 + 3x

Step 4 — Group like terms. The

x

terms are

10x

-8x

, and

3x

. The constants are

-15

and

14

(10x - 8x + 3x) + (-15 + 14)

Step 5 — Add the coefficients in each group. For the variable terms:

10 - 8 + 3 = 5

, giving

5x

. For the constants:

-15 + 14 = -1

5x - 1

Step 6 — Check by substitution. Let

x = 2

. Original:

5(4-3) - 2(8-7) + 6 = 5(1) - 2(1) + 6 = 5 - 2 + 6 = 9

. Simplified:

5(2) - 1 = 9

. They match, so

5x - 1

is correct.

Practice questions

Simplify

-3(2x - 5) + 4x

$-2x + 15$
$-2x - 15$
$-10x + 15$
$-2x + 5$

Answer: $-2x + 15$

Distribute

-3

across both terms:

(-3)(2x) = -6x

and

(-3)(-5) = +15

, giving

-6x + 15 + 4x

. Combine the

x

terms:

-6x + 4x = -2x

. The result is

-2x + 15

. The answer

-2x - 15

comes from forgetting that a negative times a negative is positive, and

-10x + 15

comes from combining

-6x

with

4x

incorrectly as though the signs were both negative.

A student simplified

12 - (5x - 8)

and got

4 - 5x

. Explain the mistake and give the correct simplified expression.

Answer: The correct answer is $20 - 5x$ (equivalently $-5x + 20$ ). The student distributed the minus sign only to the first term inside the parentheses.

The minus sign in front of the parentheses means multiply by

-1

12 - 1(5x - 8) = 12 - 5x + 8

. Both terms inside change sign, so

-8

becomes

+8

. Combining constants gives

12 + 8 = 20

, so the expression simplifies to

20 - 5x

. The student's version,

4 - 5x

, came from computing

12 - 8

, which treats the

-8

as if it stayed negative. A quick check with

x = 0

settles it: the original is

12 - (-8) = 20

, matching

20 - 5x

, not

4

Simplify

4(3y + 2) - (y - 6) + y^2

and state how many terms your answer has.

Answer: $y^2 + 11y + 14$ , which has three terms.

Distribute the

4

12y + 8

. Distribute the invisible

-1

-y + 6

. The expression becomes

12y + 8 - y + 6 + y^2

. Combine the

y

terms:

12y - y = 11y

(remember

-y

has coefficient

-1

). Combine constants:

8 + 6 = 14

. The

y^2

term has no partner, since

y^2

and

y

are not like terms. Written in descending order of exponent, the answer is

y^2 + 11y + 14

— three terms.

FAQ

Are $3x$ and $3x^2$ like terms?: No. Like terms must match in both the variable and the exponent. $3x$ counts groups of $x$ , while $3x^2$ counts groups of $x \cdot x$ , so they cannot be added into one term. You can verify this: at $x = 4$ , $3x = 12$ but $3x^2 = 48$ . The expression $3x + 3x^2$ is already simplified.
Why do all the signs flip when I subtract a parenthesis?: Because a bare minus sign in front of parentheses is really a multiplier of $-1$ , and that $-1$ distributes to every term inside. So $-(2x - 9)$ becomes $(-1)(2x) + (-1)(-9) = -2x + 9$ . Writing the $-1$ in explicitly before you distribute prevents the most common error in this lesson.
Do I always have to distribute first?: Distribute before combining anything that sits outside the parentheses with something inside — that part is required by the order of operations. But if the terms inside the parentheses are themselves like terms, you may simplify inside first. In $3(2x + 5x)$ , you can combine to get $3(7x) = 21x$ , or distribute to get $6x + 15x = 21x$ . Both routes give the same result.
How do I know my simplified expression is correct?: Substitute a convenient number for the variable — $x = 2$ or $x = 3$ works well, but avoid $x = 0$ and $x = 1$ since they can hide sign errors — and evaluate both the original expression and your answer. Equivalent expressions must give the same value. If the two values differ, retrace your distribution step first; a dropped negative is the usual cause.

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