Can you please help me with this one?

Question

(x³ + ax² + bx + c)/ (x+d) = x² - kx + k

In the above equation d and k are positive integers. Which of the following could be all of the roots of x³ + ax² + bx + c?

1. (A) d, -k
2. (B) d, k
3. (C) -d, 2
4. (D) -1, -2

Answer 1

Hi Sona

This is an interesting problem. Let's start by multiply the original equation by x+ d

We now have x^3+ax^2 + bx + c = (x+d) (x^2 -kx +k) . Now to find the roots of the left had side of this equation, we set the right side equal to zero.

From these two factors, we can immediately see that (x+d) = 0 yields x- -d as one of the roots. (Since there's only one answer that has -d as a root, we could pick (C) as the answer) You'd probably do that in the ACT test, but let's explore why that's correct.

Using the quadratic theorem on the equation (x^2 -kx +k) gives us two roots x = (k +- Sqrt(k^2-4k))/2

But, even though x^3+ax^2 + bx + c has three roots, the answer choices only show two root. This means that the two roots of x^2 -kx +k must be the same - a double root.

A double root exists when the discriminant is equal to 0. But k^2-4k is zero when k=4.So the double roots occurs when k-4 or x = 4/2 = 2

So the answer is indeed (C) >>>> -d, 2

Hope this helps.

Answer 2

-d is already a root.

Which option has a -d?