Question

what expression does this set of algebra tiles represent?
combine like terms. for example, write 3 instead of 1 + 1 + 1.

Explanation:

Step1: Count \( x^2 \) tiles

There are 7 \( x^2 \) tiles (4 in first row, 3 in second row). So term is \( 7x^2 \).

Step2: Count \( x \) tiles

There is 1 \( x \) tile. So term is \( x \).

Step3: Count 1 tiles

Left yellow: \( 1 + 1 + 1 + 1 = 4 \) (2 in first row, 2 in next two rows? Wait, left yellow: first row 2, second row 1, third row 1? Wait no, looking at the image: left yellow has 2 (first row), 1 (second), 1 (third)? Wait no, the left yellow: first row (below \( x^2 \) tiles) has two 1s, then below that two rows with one 1 each? Wait no, the left yellow: let's recount. The yellow 1s: left side: two in first row (under \( x^2 \) tiles), then one below, one below that? Wait no, the image: left yellow: first row (horizontal) two 1s, then two vertical rows (each with one 1) below? Wait no, the right side (next to \( x \) tile) has 3 rows of two 1s? Wait no, the right yellow: 3 rows, each with two 1s: 3*2=6. The left yellow: two 1s (first row) and two 1s (next two rows? Wait no, the left yellow: first row (under \( x^2 \) tiles) has two 1s, then below that, two rows with one 1 each? Wait no, the left yellow: let's count all 1s. Left: 2 (top) + 1 + 1 = 4? Wait no, the left yellow: first row (horizontal) two 1s, then two vertical (each one 1) below: 2 + 1 + 1 = 4? Wait no, the right yellow (next to \( x \) tile): 3 rows, each with two 1s: 3*2=6. So total 1s: 4 + 6 = 10? Wait no, let's look again. The left yellow: below the \( x^2 \) tiles (the two \( x^2 \) tiles on the left? Wait no, the \( x^2 \) tiles are 7: first row 4, second row 3. Then, to the right of the second row \( x^2 \) tiles: a green \( x \) tile, then yellow 1s: 3 rows, each with two 1s (so 3*2=6). Then, below the first two \( x^2 \) tiles (leftmost) there are yellow 1s: first row (horizontal) two 1s, then below that, two rows (each with one 1) so 2 + 1 + 1 = 4? Wait no, the left yellow: first row (under \( x^2 \) tiles) has two 1s, then two rows (each with one 1) below: 2 + 1 + 1 = 4. The right yellow: 3 rows, each with two 1s: 3*2=6. So total 1s: 4 + 6 = 10? Wait no, that can't be. Wait the problem says "combine like terms". Let's do it step by step.

Wait the \( x^2 \) tiles: 7 (4 in first row, 3 in second: 4+3=7). So \( 7x^2 \).

The \( x \) tiles: 1 (the green one). So \( x \).

The 1 tiles: let's count all yellow 1s. Left side: two 1s (top) and two 1s (below: one in second row, one in third row? Wait no, the left yellow: first row (horizontal) two 1s, then two vertical (each one 1) below: 2 + 1 + 1 = 4. The right yellow (next to \( x \) tile): 3 rows, each with two 1s: 3*2=6. So total 1s: 4 + 6 = 10? Wait no, 4 + 6 = 10? Wait 2 (left top) + 1 + 1 (left bottom) = 4, and right: 2*3=6 (3 rows, 2 each). So 4 + 6 = 10. So total 1s: 10.

Wait but let's check again. The left yellow: first row (under \( x^2 \) tiles) has two 1s, then below that, two rows (each with one 1) so 2 + 1 + 1 = 4. The right yellow: next to the \( x \) tile, there are 3 rows, each with two 1s: 3*2=6. So 4 + 6 = 10. So total 1s: 10.

So combining like terms: \( 7x^2 + x + 10 \). Wait no, wait the left yellow: maybe I miscounted. Let's see: the left yellow: two 1s (top) and two 1s (bottom two rows: each one 1) so 2 + 1 + 1 = 4. The right yellow: 3 rows, each with two 1s: 3*2=6. So 4 + 6 = 10. So total 1s: 10. So the expression is \( 7x^2 + x + 10 \). Wait but let's check the \( x^2 \) tiles: first row 4, second row 3: 4+3=7. Correct. \( x \) tiles: 1. Correct. 1s: 4 + 6 = 10. Correct. So the expression is \( 7x^2 + x + 10 \).

Wait but maybe I made a mistake in counting the 1s. Let's re-express:

- \( x^2 \) tiles: 7 (4 in top row, 3 in bottom row: 4 + 3 = 7) → \( 7x^2 \)
- \( x \) tiles: 1 (the green one) → \( x \)
- Constant (1) tiles:
- Left side: 2 (top) + 1 + 1 = 4 (wait, no, the left side: two 1s (horizontal) and two 1s (vertical) → 2 + 2 = 4? Wait, maybe the left yellow is two 1s (first row) and two 1s (next two rows, each one 1) → 2 + 1 + 1 = 4.
- Right side (next to \( x \) tile): 3 rows, each with two 1s → 3 * 2 = 6.
- Total constants: 4 + 6 = 10 → \( 10 \)

So combining like terms: \( 7x^2 + x + 10 \).

Answer:

\( 7x^2 + x + 10 \)