Question

a polynomial expression is shown.
$6x^2 - 7x - 24$
which pair of binomials are factors of $6x^2 - 7x - 24?$
$\square\\ 6x + 4$
$\square\\ 2x + 3$
$\square\\ 3x - 6$
$\square\\ 2x - 3$
$\square\\ 3x - 8$
$\square\\ 2x - 4$
$\square\\ 2x + 4$
$\square\\ x - 6$

Explanation:

Step1: Recall factoring quadratic trinomial \(ax^2 + bx + c\) (here \(a = 6\), \(b=-7\), \(c = - 24\)). We need two numbers \(m\) and \(n\) such that \(m\times n=a\times c=6\times(-24)=-144\) and \(m + n=b=-7\).

Find two numbers: \(m = 9\), \(n=-16\) (since \(9\times(-16)=-144\) and \(9+(-16)=-7\)).

Step2: Rewrite the middle term using \(m\) and \(n\):

\(6x^{2}+9x - 16x-24\)

Step3: Group the terms:

\((6x^{2}+9x)+(-16x - 24)\)

Step4: Factor out GCF from each group:

\(3x(2x + 3)-8(2x + 3)\)

Step5: Factor out the common binomial \((2x + 3)\):

\((3x - 8)(2x + 3)\)
So the pair of binomials that are factors are \(2x + 3\) and \(3x - 8\).

Answer:

2x + 3, 3x - 8