Question

the diagram represents $6x^2 - 7x + 2$ with a factor of $2x - 1$. what is the other factor of $6x^2 - 7x + 2$? $\bigcirc\\ 3x - 2$ $\bigcirc\\ 3x - 1$ $\bigcirc\\ 3x + 1$ $\bigcirc\\ 3x + 2$ (and the chart with cells: 2x, -1;?, 6x², -3x;?, -4x, 2)

Explanation:

Step1: Recall factoring quadratic

To factor \(6x^2 - 7x + 2\) with a known factor \(2x - 1\), we can use polynomial division or reverse the area model. The area model's rows and columns represent factors. The first column's top is \(2x\), and the middle cell is \(6x^2\), so the first element of the other factor (let's call it \(ax + b\)) satisfies \(2x \times ax = 6x^2\). Solving \(2a = 6\) gives \(a = 3\).

Step2: Find the constant term

Now, the last cell of the area model is \(2\), and the top - right cell is \(-1\). So \( - 1\times b=2\)? Wait, no, looking at the bottom row: the middle term is \(-4x\) and the right term is \(2\). The known factor is \(2x - 1\), and we found the first part of the other factor is \(3x\). Let's multiply \((2x - 1)(3x + c)\) and expand: \(6x^2+2cx-3x - c=6x^2+(2c - 3)x - c\). Comparing with \(6x^2-7x + 2\), we have \(-c = 2\) (so \(c=-2\)) and \(2c-3=-7\) (substitute \(c = - 2\): \(2\times(-2)-3=-4 - 3=-7\), which matches). So the other factor is \(3x - 2\).

Answer:

\(3x - 2\)