Adding and Subtracting Polynomials Using Algebra Tiles

Adding and subtracting polynomials using algebra tiles means building each polynomial with tiles, then combining like tiles. To add, put all the tiles together and remove zero pairs. To subtract, either flip every tile of the second polynomial and add, or take its tiles away, adding zero pairs when you are short. So \((x^2 + x + 2) + (3x^2 – 2x + 2) = 4x^2 – x + 4\).

Build each polynomial as you read: Try it with our free Algebra Tiles tool. Its tile mat flips tiles and clears zero pairs for you.

The tiles you need

• x²-tile: a large square, x long on each side, worth \(x^2\).
• x-tile: a long rectangle, x long and 1 wide, worth \(x\).
• Unit tile: a small square worth \(1\).

In our pictures positive tiles are white and shaded (or red) tiles are negative. A positive and a negative tile of the same shape make a zero pair worth 0. Only tiles of the same shape combine: an x²-tile and an x-tile never cancel, which is exactly why you can only combine like terms.

Step by step: add (x² + x + 2) + (3x² − 2x + 2)

We will add \(x^2 + x + 2\) and \(3x^2 – 2x + 2\). The first polynomial is 1 x²-tile, 1 x-tile and 2 unit tiles, all positive. The second is 3 x²-tiles, 2 shaded (negative) x-tiles and 2 positive unit tiles.

1. Model each polynomial. Lay out the tiles for each polynomial in its own row.

   

2. Group like tiles. Push the x²-tiles together, the x-tiles together and the unit tiles together. You have 4 x²-tiles, 1 positive x-tile with 2 negative x-tiles, and 4 unit tiles.
3. Remove zero pairs. The positive x-tile and one negative x-tile cancel. Cross them out.

   

4. Read what is left. There are 4 x²-tiles, 1 negative x-tile and 4 unit tiles: \(4x^2 – x + 4\).
5. Check with symbols. \(1 + 3 = 4\) x²-tiles, \(1 + (-2) = -1\) x-tiles, \(2 + 2 = 4\) unit tiles. The result matches.
6. Try it. Open the free Algebra Tiles tool, put both polynomials on the tile mat and let it clear the zero pair. You should end with the same \(4x^2 – x + 4\).

Subtracting polynomials with algebra tiles: two ways

Subtraction is where tiles help most, because they show why the signs change. We will find \((2x^2 + 3x – 1) – (x^2 – 2x + 4)\) both ways.

Way 1: add the opposite

1. Model the first polynomial: 2 x²-tiles, 3 x-tiles and 1 shaded unit tile.
2. Model the second polynomial, \(x^2 – 2x + 4\): 1 x²-tile, 2 shaded x-tiles and 4 unit tiles.
3. Subtracting is adding the opposite, so flip every tile of the second polynomial. It becomes 1 shaded x²-tile, 2 positive x-tiles and 4 shaded unit tiles: \(-x^2 + 2x – 4\).
4. Combine with the first polynomial and remove zero pairs. One x²-tile pair cancels, leaving 1 x²-tile. The x-tiles are all positive: \(3 + 2 = 5\). The units are all negative: \(1 + 4 = 5\) shaded units.
5. Result: \(x^2 + 5x – 5\).

Way 2: take away, adding zero pairs when you are short

1. Start with only the first polynomial: 2 x²-tiles, 3 positive x-tiles, 1 shaded unit.
2. Take away 1 x²-tile. One x²-tile is left.
3. Take away 2 negative x-tiles. There are none, so add 2 zero pairs of x-tiles (2 positive, 2 negative). Now there are 5 positive and 2 negative x-tiles; take away the 2 negative ones. Five positive x-tiles are left.
4. Take away 4 positive units. There are none, so add 4 zero pairs of units. Now there are 4 positive and 5 shaded units; take away the 4 positive ones. Five shaded units are left.
5. Result: \(x^2 + 5x – 5\), the same as Way 1.

Way 2 explains Way 1: adding zero pairs and taking tiles away always leaves the opposite of what you removed. Once that clicks, flipping the second polynomial is the faster habit.

More worked examples

Example 1: adding with negatives, (−2x² + 4x − 3) + (x² − 6x + 5)

The first polynomial is 2 shaded x²-tiles, 4 x-tiles and 3 shaded units. The second is 1 x²-tile, 6 shaded x-tiles and 5 units.

• x²-tiles: 1 positive cancels 1 of the 2 shaded ones, leaving 1 shaded x²-tile.
• x-tiles: 4 positive cancel 4 of the 6 shaded ones, leaving 2 shaded x-tiles.
• Units: 3 shaded cancel 3 of the 5 positive, leaving 2 positive units.

Result: \(-x^2 – 2x + 2\).

Example 2: subtracting, (x² − 3x + 2) − (2x² + x − 4)

Flip the second polynomial: \(-2x^2 – x + 4\), which is 2 shaded x²-tiles, 1 shaded x-tile and 4 units. Add it to 1 x²-tile, 3 shaded x-tiles and 2 units.

• x²-tiles: 1 positive and 2 shaded leave 1 shaded: \(-x^2\).
• x-tiles: 3 shaded and 1 shaded make 4 shaded: \(-4x\).
• Units: 2 and 4 make 6: \(+6\).

Result: \(-x^2 – 4x + 6\).

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Common mistakes when adding and subtracting polynomials

• Flipping only the first tile. In \((2x^2 + 3x – 1) – (x^2 – 2x + 4)\), every tile of the second polynomial flips, not just the x²-tile. Flipping only one gives \(x^2 + 3x – 1 – 2x + 4 = x^2 + x + 3\), which is wrong.
• Cancelling unlike tiles. A positive x-tile and a shaded unit tile are not a zero pair. Only matching shapes cancel.
• Losing a zero pair. Cross out each pair as you remove it, as in the second picture. Removing one tile of a pair without its partner changes the value.
• Misreading shaded tiles. In our images the gray tiles are negative. Count them as negative before you combine anything.
• Treating x² and x as the same. \(4x^2 – x\) does not simplify to \(3x\). The tiles are different shapes because the terms are different.

Practice exercises with answers

Model each problem with tiles, combine, and remove zero pairs.

1. \((x^2 + 2x + 3) + (2x^2 + x + 1)\)
2. \((3x^2 – x + 2) + (-x^2 – 2x – 5)\)
3. \((2x^2 + 4x) + (x^2 – 4x + 3)\)
4. \((4x^2 + 3x + 5) – (x^2 + x + 2)\)
5. \((x^2 – 3x + 2) – (2x^2 + x – 4)\)
6. \((5x – 2) – (-x^2 + 2x – 3)\)

Answers:

1. \(3x^2 + 3x + 4\)
2. \(2x^2 – 3x – 3\)
3. \(3x^2 + 3\) (the 4 positive and 4 shaded x-tiles are all zero pairs)
4. \(3x^2 + 2x + 3\)
5. \(-x^2 – 4x + 6\) (Example 2)
6. \(x^2 + 3x + 1\) (flipping the shaded x²-tile makes it positive)

Adding and subtracting polynomials with tiles practice quiz: 10 questions

Test yourself with 10 multiple-choice questions. Choose an answer and press Submit to see if you are right and why. Your score appears at the end.

Frequently asked questions

How do you add polynomials with algebra tiles?

Model each polynomial with x²-tiles, x-tiles and unit tiles, using shaded tiles for negative terms. Put all the tiles together, group them by shape, and remove every zero pair. The tiles left over are the sum. For \((x^2 + x + 2) + (3x^2 - 2x + 2)\) you get \(4x^2 - x + 4\).

How do you subtract polynomials with algebra tiles?

The quick way is to flip every tile of the polynomial being subtracted, then add as usual. The other way is to take its tiles away from the first polynomial, adding zero pairs whenever you do not have enough of a tile to remove. Both give the same answer.

Why do you flip the tiles when you subtract?

Subtracting a polynomial is the same as adding its opposite, and flipping a tile turns it into its opposite. The take-away method shows this: when you add zero pairs and remove one tile of each, the opposite tile stays behind.

What is a zero pair in algebra tiles?

A zero pair is one positive tile and one negative tile of the same shape, such as an x-tile and a shaded x-tile. Together they are worth 0, so you can remove a zero pair or add one without changing the value of an expression.

Can an x²-tile cancel an x-tile?

No. Tiles only cancel when they are the same shape, because \(x^2\) and \(x\) are different quantities. That is the tile version of the rule that you can only combine like terms.

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