Question

Which of the following could be used as justification that 3x^2+10x-8 is not prime over the set of rational numbers

A. (3x+2)(x-4)
B. (3x+4)(x-2)
C. (3x-2)(x+4)
D. (3x-4)(x+2)

Answer 1

C

Explanation:

Consider the trinomial
`3x^2+10x-8.`

We can rewrite it as

`3x^2+10x-8=3x^2+12x-2x-8.`

Now group first two terms and last two terms:

`(3x^2+12x)+(-2x-8).`

The common factor in first two terms is
`3x` and the common factor in last two terms is
`-2.` Use the distributive property for both groups of terms:

`3x^2+12x=3x(x+4),\\ \\(-2x-8)=-2(x+4),`

so

`(3x^2+12x)+(-2x-8)=3x(x+4)-2(x+4).`

Now you can see that
`(x+4)` is a common factor, thus

`3x(x+4)-2(x+4)=(x+4)(3x-2)=(3x-2)(x+4).`

Since the trinomial can be represented as a product of binomials, this trinomial is not prime.