Question

which polynomial is represented by the algebra tiles? \\( \circ \\ x^2 - x - 4 \\) \\( \circ \\ x^2 - x + 4 \\) \\( \circ \\ 3x^2 - 5x + 8 \\) \\( \circ \\ 3x^2 - 5x - 8 \\)

Explanation:

Step1: Count \(x^2\) terms

We have two \(+x^2\) tiles and one \(-x^2\) tile. So the total for \(x^2\) terms is \(2x^2 - x^2 = x^2\)? Wait, no, wait: Wait, two \(+x^2\) and one \(-x^2\)? Wait, no, looking at the tiles: first row: two orange \(+x^2\), one blue \(-x^2\). Wait, no, wait the first row: first tile \(+x^2\), second \(+x^2\), third \(-x^2\). So that's \(2x^2 - x^2 = x^2\)? Wait, no, wait maybe I miscounted. Wait, no, the first row: two \(+x^2\) (orange) and one \(-x^2\) (blue). So \(2x^2 - x^2 = x^2\)? Wait, no, wait the second row: wait, no, let's re-examine. Wait, the \(x^2\) tiles: two \(+x^2\) (orange) and one \(-x^2\) (blue). So total \(x^2\) terms: \(2x^2 - x^2 = x^2\)? Wait, no, maybe I made a mistake. Wait, no, the first row: three tiles? Wait, first row: three tiles: two orange \(+x^2\), one blue \(-x^2\). So \(2x^2 - x^2 = x^2\)? Wait, no, wait the second row: the \(x\) terms. Let's check the \(x\) terms. The blue tiles are \(-x\), orange are \(+x\). So blue \(-x\) tiles: three? Wait, first row of \(x\) tiles: two blue \(-x\), one orange \(+x\). Second row of \(x\) tiles: one blue \(-x\), one orange \(+x\). So total \(-x\) tiles: \(2 + 1 = 3\), total \(+x\) tiles: \(1 + 1 = 2\). So total \(x\) terms: \(2x - 3x = -x\). Now the constant terms: blue tiles are \(-1\) (since they are negative units), orange are \(+1\). Blue \(-1\) tiles: two (in the third and fourth rows? Wait, third row: one blue \(-1\), three orange \(+1\). Fourth row: one blue \(-1\), three orange \(+1\). So total blue \(-1\) tiles: \(2\), total orange \(+1\) tiles: \(3 + 3 = 6\). So constant terms: \(6 - 2 = 4\). So putting it all together: \(x^2 - x + 4\). Wait, let's verify again.

Wait, \(x^2\) terms: two \(+x^2\) (orange) and one \(-x^2\) (blue). So \(2x^2 - x^2 = x^2\). \(x\) terms: blue \(-x\) tiles: three (first row of \(x\): two, second row of \(x\): one; total three \(-x\)), orange \(+x\) tiles: two (first row of \(x\): one, second row of \(x\): one; total two \(+x\)). So \(2x - 3x = -x\). Constant terms: blue \(-1\) tiles: two (third row: one, fourth row: one), orange \(+1\) tiles: six (third row: three, fourth row: three). So \(6 - 2 = 4\). So the polynomial is \(x^2 - x + 4\). Let's check the options. The options are \(x^2 - x - 4\), \(x^2 - x + 4\), \(3x^2 -5x +8\), \(3x^2 -5x -8\). So the correct one is \(x^2 - x + 4\), which is the second option.

Wait, but wait, maybe I miscounted the \(x^2\) terms. Wait, first row: three tiles: two \(+x^2\) (orange) and one \(-x^2\) (blue). So \(2x^2 - x^2 = x^2\). Then \(x\) terms: three \(-x\) (blue) and two \(+x\) (orange), so \(-x\). Constants: two \(-1\) (blue) and six \(+1\) (orange), so \(+4\). So the polynomial is \(x^2 - x + 4\), which is the second option.

Step2: Verify with each option

Let's check each option:

1. \(x^2 - x - 4\): constant term is \(-4\), but we have \(+4\), so no.

1. \(x^2 - x + 4\): matches our calculation: \(x^2\) term, \(-x\) term, \(+4\) constant.

1. \(3x^2 -5x +8\): \(x^2\) term is \(3x^2\), but we have \(x^2\), so no.

1. \(3x^2 -5x -8\): \(x^2\) term \(3x^2\), constant \(-8\), no.

So the correct option is \(x^2 - x + 4\).

Answer:

\(x^2 - x + 4\) (the second option, i.e., the option with \(x^2 - x + 4\))