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

A quadratic \(ax^2 + bx + c\) with a factor \((mx + n)\) can be factored as \((mx + n)(px + q)\) where \(mp=a\), \(nq = c\), and \(mq+np = b\). Here, \(a = 3\), \(b=-10\), \(c = 8\), and one factor is \(3x - 4\). Let the other factor be \(x + k\) (since \(3\times1=3\) for the \(x^2\) coefficient).

Step2: Multiply the factors

Multiply \((3x - 4)(x + k)=3x^2+3kx-4x - 4k=3x^2+(3k - 4)x-4k\).

Step3: Compare coefficients

Compare with \(3x^2-10x + 8\). So, \(-4k = 8\) (constant term) and \(3k - 4=-10\) (linear term). Solve \(-4k = 8\) gives \(k=-2\). Wait, no, wait: Wait, if we set the other factor as \(x - 2\), then \((3x - 4)(x - 2)=3x^2-6x-4x + 8=3x^2-10x + 8\), which matches. Let's check the constant term: \(-4\times(-2)=8\), correct. Linear term: \(3\times(-2)-4=-6 - 4=-10\), correct.

Answer:

A. \(x - 2\)