Question

1. which expression is a factor of ( 30x^2 - 4x - 16 )?

options:
( 5x - 4 )
( 3x - 4 )
( 3x - 2 )
( 5x + 4 )

Explanation:

Step1: Factor the quadratic expression

First, we can try to factor the quadratic expression \(30x^{2}-4x - 16\). We can factor out a common factor of 2 first:
\(30x^{2}-4x - 16=2(15x^{2}-2x - 8)\)

Now, we need to factor the quadratic \(15x^{2}-2x - 8\). We look for two numbers \(a\) and \(b\) such that \(a\times b=15\times(- 8)=-120\) and \(a + b=-2\). After some trial and error, we find that \(a = -12\) and \(b = 10\) since \(-12\times10=-120\) and \(-12 + 10=-2\).

We rewrite the middle term using these two numbers:
\(15x^{2}-12x+10x - 8\)

Then we group the terms:
\((15x^{2}-12x)+(10x - 8)\)

Factor out the common factors from each group:
\(3x(5x - 4)+2(5x - 4)\)

Now we can factor out \((5x - 4)\):
\((3x + 2)(5x - 4)\)

So the original expression \(30x^{2}-4x - 16=2(3x + 2)(5x - 4)\)

Step2: Identify the factor

From the factored form, we can see that \(5x - 4\) is a factor of \(30x^{2}-4x - 16\)

Answer:

A. \(5x - 4\)