Question

the model of a trinomial is shown. what are the factors of the trinomial? select two options. +x² -x -x -x -x -x -x -x +x - - - - - - - +x - - - - - - - □x - 14 □x + 7 □x - 7 □x - 2 □x + 2

Explanation:

Step1: Determine the trinomial

From the model, the trinomial is \(x^{2}-7x + 2x\)? Wait, no. Wait, the orange part is \(+x^{2}\), then the blue parts: first row of blue is \(-x\) repeated 7 times, so \(-7x\), then the next two rows (the ones with \(-\)): wait, no, the left column (orange) has \(+x^{2}\), \(+x\), \(+x\). Wait, no, let's re - examine. The area model: the top - left is \(x^{2}\), then to the right of \(x^{2}\) are 7 rectangles of \(-x\), so that's \(-7x\). Then below \(x^{2}\) are two rectangles of \(+x\) (wait, no, the left column below \(x^{2}\) is \(+x\) and \(+x\)? Wait, no, the left column (vertical) and top row (horizontal) form the factors. Wait, the vertical side: let's see the left - most column (the one with the orange and the two \(+x\)s). The vertical terms: the first term is \(x\) (from \(x^{2}=x\times x\)? Wait, no. Wait, the trinomial is formed by adding all the terms. The orange square is \(x^{2}\), the first row of blue rectangles: 7 of \(-x\), so \(-7x\), the second and third rows (the ones with \(-\) symbols? Wait, no, the second and third rows (below the first blue row) are \(+x\) and \(+x\) on the left, and then 7 of \(-1\) each? Wait, no, maybe I misread. Wait, the correct way: the area model for factoring \(ax^{2}+bx + c\) is a rectangle with length and width as the factors. Let's sum up all the terms:

The top - left: \(x^{2}\)

To the right of \(x^{2}\): 7 rectangles of \(-x\), so \(-7x\)

Below \(x^{2}\): two rectangles of \(+x\) (so \(+x + x=+2x\))

And below the \(-7x\) rectangles: 7 rectangles of \(-1\) each? Wait, no, that can't be. Wait, maybe the trinomial is \(x^{2}-7x + 2x\)? No, that's \(x^{2}-5x\), which is wrong. Wait, no, let's look again. The left column (the vertical part) has three terms: \(x\), \(+1\), \(+1\)? Wait, no, the correct way is: the trinomial is \(x^{2}-7x + 14\)? No, wait, the vertical side (let's call it the height) and horizontal side (length) are the factors. Let's find the trinomial by adding all the areas.

The orange square: \(x^{2}\)

The first row of blue rectangles (horizontal, to the right of \(x^{2}\)): 7 rectangles with area \(-x\), so total area \(-7x\)

The second row (below the first blue row) has a rectangle with area \(+x\) (on the left) and 7 rectangles with area \(-1\), so \(+x-7\)

The third row (below the second row) has a rectangle with area \(+x\) (on the left) and 7 rectangles with area \(-1\), so \(+x - 7\)

Wait, no, this is getting confusing. Let's use the standard area - model factoring. The trinomial is \(x^{2}-7x + 14\)? No, that's not right. Wait, maybe the trinomial is \(x^{2}-5x + 14\)? No. Wait, let's start over.

The area model: the top - left term is \(x^{2}\), which is \(x\times x\). Then, the number of \(-x\) terms: 7, so that's \(-7x\). Then the constant terms: let's see, below the \(x^{2}\) we have two \(+x\) terms? No, wait, the vertical factor: let's say the vertical side is \((x + 2)\) and the horizontal side is \((x-7)\). Let's multiply \((x + 2)(x - 7)=x^{2}-7x+2x - 14=x^{2}-5x - 14\). Wait, no, that's not matching. Wait, maybe the vertical side is \((x - 7)\) and the horizontal side is \((x + 2)\)? No. Wait, another approach: the trinomial is \(x^{2}-7x + 2x\)? No, that's \(x^{2}-5x\). Wait, I think I made a mistake in reading the model. Let's look at the vertical and horizontal factors.

The left - most column (vertical) has terms: \(x\), \(+1\), \(+1\)? Wait, no, the vertical factor: the first term is \(x\) (from \(x^{2}=x\times x\)), and then two \(+1\)s? No, the two \(+x\)s: wait, the left column (the one with the orange \(x^{2}\), \(+x\), \(+x\)): so the vertical factor is \(x + 2\). The top row (horizontal) has \(x\) and \(-7\), so the horizontal factor is \(x-7\). Let's multiply \((x + 2)(x - 7)=x^{2}-7x+2x - 14=x^{2}-5x - 14\). Wait, but the model: the \(x^{2}\), then 7 of \(-x\) (so \(-7x\)), then 2 of \(+x\) (so \(+2x\)), and then \(2\times(-7)=-14\). So the trinomial is \(x^{2}-7x + 2x-14=x^{2}-5x - 14\). Now, let's factor \(x^{2}-5x - 14\). We need two numbers that multiply to \(-14\) and add up to \(-5\). The numbers are \(-7\) and \(2\). So \(x^{2}-5x - 14=(x + 2)(x - 7)\). Wait, but the options are \(x - 14\), \(x + 7\), \(x - 7\), \(x - 2\), \(x + 2\). Wait, maybe I misread the model. Let's try again.

Wait, the orange square is \(x^{2}\), the first row of blue rectangles: 7 of \(-x\) (so \(-7x\)), the second row (below \(x^{2}\)): 1 of \(+x\), the third row (below the second row): 1 of \(+x\), and then the area below the \(-7x\) rectangles: 7 of \(-2\)? No, this is getting too confusing. Let's use the fact that in the area model, the sum of the terms is the trinomial. The trinomial is \(x^{2}-7x+2x - 14=x^{2}-5x - 14\). Wait, no, maybe the trinomial is \(x^{2}-7x + 2x\)? No, that's \(x^{2}-5x\), which is a binomial. Wait, I think the correct trinomial from the model is \(x^{2}-7x + 2x-14\)? No, let's look at the vertical and horizontal sides. The vertical side has \(x\), \(+1\), \(+1\) (so \(x + 2\)) and the horizontal side has \(x\), \(-7\) (so \(x - 7\)). So the factors are \(x + 2\) and \(x - 7\). But wait, the options are \(x-14\), \(x + 7\), \(x - 7\), \(x - 2\), \(x + 2\). Wait, maybe the trinomial is \(x^{2}-5x - 14\), and the factors are \(x + 2\) and \(x - 7\). So among the options, \(x + 2\) and \(x - 7\) are the factors. Wait, but let's check again.

Wait, another way: let's assume the trinomial is \(x^{2}-7x+2x - 14\) is wrong. Let's look at the number of \(-x\) terms: 7, so the coefficient of \(x\) is \(-7\) plus the coefficient from the lower part. The lower part (the two \(+x\)s) gives \(+2x\), so total coefficient of \(x\) is \(-7 + 2=-5\), and the constant term is \(2\times(-7)=-14\). So the trinomial is \(x^{2}-5x - 14=(x + 2)(x - 7)\). So the factors are \(x + 2\) and \(x - 7\).

Step2: Match with the options

Among the given options: \(x - 7\) and \(x + 2\) are the factors. Wait, but let's check the multiplication of \(x + 2\) and \(x - 7\):

\((x + 2)(x - 7)=x\times x+x\times(-7)+2\times x+2\times(-7)=x^{2}-7x + 2x-14=x^{2}-5x - 14\)

Now, let's check the other options:

- \((x - 14)(x + a)=x^{2}+(a - 14)x-14a\), which is not our trinomial.
- \((x + 7)(x + b)=x^{2}+(b + 7)x + 7b\), not our trinomial.
- \((x - 7)(x + 2)=x^{2}-5x - 14\), which matches.
- \((x - 2)(x + c)=x^{2}+(c - 2)x-2c\), not our trinomial.

Answer:

\(x - 7\), \(x + 2\) (i.e., the options are: \(x - 7\) and \(x + 2\))