Grade by Grade
C. Algebra · Algebra: Expressions, Equations and Coding

Collecting Like Terms: Adding and Subtracting Polynomials

C1.4 A1About 70 minutes
Learning goals
  • I can identify like terms, including terms with x² and with two variables.
  • I can simplify an expression by collecting like terms, using algebra tiles and symbols.
  • I can add polynomials and subtract polynomials by adding the opposite.
  • I can check a simplification by substituting a value.

Words to know

like terms
Terms with exactly the same variables raised to the same exponents, such as 4x² and −x², or 3ab and 5ba. Only their coefficients may differ.
unlike terms
Terms whose variable parts differ, such as 3x and 3x², or 2x and 2y. They cannot be combined into one term.
zero pair
A positive tile and a matching negative tile. Together they equal 0, so they can be removed.
simplify
Rewrite an expression in an equivalent, shorter form, for example by collecting like terms.
opposite of a polynomial
The polynomial with every term's sign changed. The opposite of x² − 4x + 1 is −x² + 4x − 1.

Remember from Grade 8

In Grade 8 you added and subtracted monomials like 5x − 2x, and added binomials with algebra tiles using zero pairs. Now you will work with x²-terms, with two variables, and you will subtract whole polynomials.

Let's start with a hot-chocolate stand

A school club runs a hot-chocolate stand at the winter carnival. For c cups, the money they take in (revenue) is 2.5c dollars. Their costs are 0.75c + 40 dollars (75¢ of supplies per cup, plus $40 to rent an urn).

Profit = revenue − cost = 2.5c − (0.75c + 40).

To make this useful, we need to simplify it. That's what this lesson is about.

Like terms

You can only combine terms that are alike: the same variables with the same exponents.

  • Like: 3x and −5x; 4x² and x²; 2ab and −7ba (ab = ba)
  • Unlike: 3x and 3x²; 2x and 2y; 5 and 5x

To collect like terms, add their coefficients and keep the variable part the same: 4x² + x² = 5x² (not 5x⁴!). Think of it like counting objects: 4 x²-tiles and 1 more x²-tile make 5 x²-tiles.

(2x² + 3x − 4)+ (x² − 5x + 2)3 zero pairs of x-tiles and 2 zero pairs of 1-tiles cancel.Result3x² − 2x − 2
Adding (2x² + 3x − 4) + (x² − 5x + 2) with tiles. Put the same kinds together, remove zero pairs, and count what is left: 3x² − 2x − 2.

Subtracting a polynomial

Subtracting is the same as adding the opposite. To subtract a polynomial in brackets, change the sign of every term inside, then collect like terms.

(3x + 2) − (x − 4) = 3x + 2 − x + 4 = 2x + 6

With tiles, “subtracting the bracket” means flipping every tile in it, then adding.

(3x + 2) − (x − 4)−flip every tile in the bracket== 2x + 6
To subtract (x − 4), flip each of its tiles: the x-tile becomes −x and the four −1 tiles become +1. Then remove the zero pair of x-tiles: 2x + 6 remains.
Worked example

Example 1: Collect like terms

Simplify 4a² − 3a + 2a² + 7a − 9.

  1. Group like terms: (4a² + 2a²) + (−3a + 7a) − 9.
  2. 4a² + 2a² = 6a². −3a + 7a = 4a.
  3. Result: 6a² + 4a − 9.
  4. Check with a = 1: original 4 − 3 + 2 + 7 − 9 = 1; simplified 6 + 4 − 9 = 1 ✓.

6a² + 4a − 9

Worked example

Example 2: Add polynomials

Add (2x² + 3x − 4) + (x² − 5x + 2).

  1. Adding: the brackets can simply be removed. 2x² + 3x − 4 + x² − 5x + 2.
  2. x²-terms: 2x² + x² = 3x². x-terms: 3x − 5x = −2x. Constants: −4 + 2 = −2.
  3. Check with x = 2: first bracket 8 + 6 − 4 = 10; second 4 − 10 + 2 = −4; total 6. Simplified: 12 − 4 − 2 = 6 ✓.

3x² − 2x − 2

Worked example

Example 3: Subtract polynomials

Simplify (3x² + 2x − 5) − (x² − 4x + 1).

  1. Add the opposite of the second polynomial: 3x² + 2x − 5 − x² + 4x − 1.
  2. x²-terms: 3x² − x² = 2x². x-terms: 2x + 4x = 6x. Constants: −5 − 1 = −6.
  3. Check with x = 1: (3 + 2 − 5) − (1 − 4 + 1) = 0 − (−2) = 2. Simplified: 2 + 6 − 6 = 2 ✓.

2x² + 6x − 6

Worked example

Example 4: Back to the hot-chocolate stand

Simplify the profit expression 2.5c − (0.75c + 40) and find the profit for 100 cups.

  1. Add the opposite: 2.5c − 0.75c − 40.
  2. Collect: 2.5c − 0.75c = 1.75c. Profit = 1.75c − 40.
  3. For c = 100: 1.75(100) − 40 = 175 − 40 = $135.
  4. Check: revenue 2.5(100) = $250; cost 0.75(100) + 40 = $115; 250 − 115 = $135 ✓.

Profit = 1.75c − 40 dollars; $135 for 100 cups.

Watch out!

  • When you subtract a bracket, the minus sign changes every term inside, not just the first one.
  • Combining like terms never changes the exponent: x + x = 2x, and x² + x² = 2x².
  • Keep the sign that is in front of each term when you move terms around.
Hands-on activity

Polynomial dice duel

You need: your homemade algebra tiles (or paper squares and strips), 2 dice, a coin, a partner.

  1. Each player builds a polynomial: roll one die for the number of x²-tiles, the other for the number of x-tiles, then roll again for the number of 1-tiles. Flip the coin for each kind: heads = positive, tails = negative. Write your polynomial down.
  2. Add the two polynomials: push all the tiles together, remove zero pairs, and write the result. Check by substituting x = 2 into the original two and the result.
  3. Subtract player 2's polynomial from player 1's: flip all of player 2's tiles, then add.
  4. The player who correctly predicts the result before building it scores a point. Play 5 rounds.

Model round: (3x² − 2x + 4) + (−x² + 5x − 6) = 2x² + 3x − 2. Subtracting instead: (3x² − 2x + 4) − (−x² + 5x − 6) = 3x² − 2x + 4 + x² − 5x + 6 = 4x² − 7x + 10.

Just for fun

Why can't x and x² ever be combined?

They're on completely different levels.

Your turn

1

Simplify 7x − 2 + 3x + 9.

2

Which of these are like terms? 4x², −x², 4x, 2y²

3

Add (3x + 5) + (2x − 8).

4

Subtract (5x − 1) − (2x + 4).

5

Add (x² + 4x − 3) + (2x² − x + 7).

6

Subtract (4x² − 3x + 2) − (x² + 2x − 6) and check with x = 1.

7

A triangle has sides 2x + 1, 3x − 2 and x + 4. Write its perimeter in simplest form.

More practice: warm-up

1

Simplify 4x + 6x.

2

Simplify 9y − 12y.

3

Simplify 3a + 2 + a.

4

Simplify x² + x².

5

Write the opposite of x − 3.

More practice: on your own

1

Simplify 6m² − 4m + 3 − 2m² + m − 8.

2

Simplify (2x² − 5x) − (−3x² + x − 4).

3

A bake sale takes in 3b dollars for b brownies. Costs are 0.8b + 25 dollars. Write and simplify the profit expression, then find the profit for 60 brownies.

4

Simplify 2ab + 3a − 5ba + 4.

5

Sam wrote (5x + 3) − (2x − 1) = 3x + 2. Find and fix his mistake.

Challenge

1

A rectangle has perimeter 10x + 6 and length 3x + 1. Find an expression for its width.

2

What polynomial must be added to 3x² − 2x + 5 to get x² + 4x − 1?

3

Three consecutive even integers can be written 2n, 2n + 2 and 2n + 4. Show that their sum is always divisible by 6.

Check your understanding

Quiz

1.

Which term is a like term with 5x²?

2.

Simplify 3x + 4 − x + 2.

3.

Simplify (2x + 3) − (x − 5).

4.

Which pair of tiles is a zero pair?

5.

Simplify (x² + 3x) + (2x² − 3x).

6.

Why can't 4x + 3y be simplified?

For parents and teachers

Big idea: C1.4 asks students to simplify algebraic expressions by applying properties of operations, using various representations and tools. Collecting like terms uses the distributive property in reverse (4x² + x² = (4 + 1)x²) and the commutative and associative properties (moving terms around).

Common difficulties:

  • Subtracting only the first term of a bracket: (5x + 3) − (2x − 1) written as 3x + 2.
  • Adding exponents when collecting: x² + x² written as x⁴ or 2x⁴.
  • Combining unlike terms: 3x + 2 written as 5x.

How to help: Play the dice duel. Ask your teen to check each answer by substituting x = 2 into both the original and the simplified expression; if they match, the simplification is probably right. Rolling dice and flipping coins makes the practice feel like a game.