What polynomial has roots of -6, -4, and 1?

Question

x^3-9x^2-22x+24x^3-x^2-26x-24x^3+x^2-26x+24x^3+9x^2+14x-24

Prof. Petra Lindgren · Accepted Answer

Write down the roots:x = -6orx = -4orx = 1Rewrite them as expressions equal to zero:x + 6 = 0orx + 4 = 0orx - 1 = 0Multiply them together to form the polynomial:f(x) = (x + 6)(x + 4)(x - 1)Now just expand that:f(x) = (x² + 10x + 24)(x - 1)f(x) = x(x² + 10x + 24) - 1(x² + 10x + 24)f(x) = x^3 + 10x² + 24x - x² - 10x - 24f(x) = x^3 + 9x² + 14x - 24Answer:D. x^3 + 9x² + 14x - 24

Prof. Petra Lindgren · Answer

I think the point here is to figure this out without resorting to the brute force method ofmutiplying out the factors. Knowing that roots of -6, -4, and 1 means that you will havefactors of (x + 6)( x + 4) and (x - 1). You don't have to do ALL the multilpications to figure outthat the constant you'd get by mutiplying out the factors is going to be -24, and this allowsyou to elminate two of the answers right away.Nice thing about having a root of 1, is that's its really easy to checkx^3-x^2-26x-24 for x = 1 is 1 - 1 - 26 - 24, and that's certainly not zero, so this one doesn'twork.The only possible answer left is the last one.

Prof. Petra Lindgren · Answer

You could multiply out the cubic (x + 6)(x + 4)(x - 1) and see which choice fits. However, there is an easier way.....The polynomial will have a first term of x^3 and a last (numerical) term of -24.Only one of the given choices fits: x^3 + 9x^2 + 14x - 24.

Abbie Toy

What polynomial has roots of -6, -4, and 1?

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