Question

`f(x)=2x^3+6x^2+4x`


`g(x)=x^2+3x+2`


The polynomials f(x) and g(x) are defined above. Which of the following polynomials is divisible by
`2x+3`?


A)
`h(x) = f(x) + g(x)`

B)
`p(x) = f(x) + 3g(x)`

C)
`r(x) = 2f(x) + 3g(x)`

D)
`s(x) = 3f(x) + 2g(x)`

Answer 1

B

Explanation:

Only answer choice B, 30, is divisible by 5, so it must be the correct answer.

p(x)=f(x)+3g(x)

Answer 2

B

Explanation:

We are given the two functions:

`f(x)=2x^3+6x^2+4x\text{ and } g(x)=x^2+3x+2`

And we want to find which of the given polynomials are divisble by (2x + 3).

First, let's factor each of the functions:

`\displaystyle \begin{aligned} f(x) &= 2x^3+6x^2+4x\\ \\ &= 2x(x^2+3x+2) \\ \\ &= 2x(x+2)(x+1)\end{aligned}`

Likewise:

`\displaystyle \begin{aligned} g(x)&=x^2+3x+2\\ \\&= (x+2)(x+1)\end{aligned}`

Let's see what happens if we add them together. This yields:

`f(x)+g(x)=2x(x+2)(x+1)+(x+2)(x+1)`

Rewriting:

`f(x)+g(x)=2x((x+2)(x+1))+1((x+2)(x+1))`

Factoring:

`f(x)+g(x)=(2x+1)((x+2)(x+1))`

Therefore, we can see that since (2x + 1) is a factor, the expression is divisible by (2x + 1).

Then to make it divisible by (2x + 1), we can multiply g by three. This yields:

`f(x)+3g(x)=2x((x+2)(x+1))+3((x+2)(x+1))`

Rewriting:

`f(x)+3g(x)=(2x+3)(x+2)(x+1)`

Since we now have a (2x + 3) term, the polynomial is now divisible by (2x + 3).

Therefore, our answer is B.