Question

the polynomial $3x^2 - 10x + 8$ has a factor of $3x - 4$. what is the other factor of $3x^2 - 10x + 8$?
$\bigcirc\\ x - 2$
$\bigcirc\\ x - 4$
$\bigcirc\\ x - 6$
$\bigcirc\\ x - 8$

Explanation:

Step1: Recall factoring of quadratic

To find the other factor, we can perform polynomial division or use factoring by grouping. Let's use the fact that if \( (3x - 4)(ax + b)=3x^{2}-10x + 8\), we can find \(a\) and \(b\). The coefficient of \(x^{2}\) is \(3\), so \(3\times a = 3\), thus \(a = 1\). Now we have \( (3x - 4)(x + b)=3x^{2}-10x + 8\). Expanding the left side: \(3x^{2}+(3b - 4)x-4b\). Comparing coefficients of \(x\): \(3b - 4=-10\), and constant term: \(-4b = 8\). From the constant term: \(-4b = 8\Rightarrow b=-2\). So the other factor is \(x - 2\).

Step2: Verify by multiplication

Multiply \( (3x - 4)(x - 2)\):

$$
LATEXBLOCK0
$$

Which matches the original polynomial.

Answer:

A. \(x - 2\) (assuming the first option is A. \(x - 2\), B. \(x - 4\), C. \(x - 6\), D. \(x - 8\))