Question

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2. Which of the following quartic functions has x=-2 and x=-3 as its only two real zeros?
A x^4-5x^3+7x^2-5x+6
B x^4+5x^3+7x^2+5x+6 ☺
C x^4+5x^3+7x^2-5x-6
D x^4-5x^3+7x^2+5x-

Answer 1

If a polynomial has those two zeros, you know it must factorise like this
\[(x+2)(x+3)\times something\]

So you have
\[(x+2)(x+3) = x^2 + 5x + 6\]

You want something with an x^4 so why not try multiplying this by x^2?


\[x^2(x+2)(x+3) = x^4 + 5x^3 + 6x^2\]

Now already we can eliminate two answers! A and D both have

\[x^4 - 5x^3 ...\]

so they are wrong.

So we're left with B or C

We need a 5x and a 6 in the answer, but it's either plus or minus 1 looking at B or C (because (x+2)(x+3) gives you a 5x and a 6, right?). So let's look at the x^2 term. It's 7x^2 for both of them and at the moment we have 6x^2 term. Now you know it has to be

\[(x^2+1)(x+2)(x+3) = x^4 + 5x^3 + 7x^2 + 5x + 6\]

and that's B