Answer: Choice B
f(x) = x^4 - 5x^3 + x^2 + 21x - 18
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Step-by-step explanation:
The root x = -2 means x+2 is a factor. This is because we add 2 to both sides of x = -2 to get x+2 = 0.
Similarly, x = 3 leads to x-3 being another factor. It's a double root so we really have two copies of (x-3)
The last root is x = 1 which gives the factor x-1.
The four factors are: (x+2)(x-3)(x-3)(x-1)
Let's use the FOIL rule to expand out the first two factors
(x+2)(x-3) = x^2-3x+2x-6 = x^2-x-6
Do the same for the last two factors
(x-3)(x-1) = x^2-1x-3x+3 = x^2-4x+3
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So far we have:
(x+2)(x-3) = x^2-x-6
(x-3)(x-1) = x^2-4x+3
which leads to
(x+2)(x-3)(x-3)(x-1) = (x^2-x-6)(x^2-4x+3)
From here I'll use the box method to multiply these trinomials. Check out the diagram below. Each inner cell is the result of multiplying the headers. Example: x^2 times x^2 = x^4 in the upper left corner.
I've color-coded the inner cells to show the like terms. Add up those like terms:
- -4x^3 + (-x^3) = -5x^3
- 3x^2 + (4x^2) + (-6x^2) = x^2
- -3x + 24x = 21x
Therefore we end up with
(x^2-x-6)(x^2-4x+3) = x^4 - 5x^3 + x^2 + 21x - 18 which is choice B
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To verify this answer, plug in x = 0 and you should get y = -18 to show that we have the correct y intercept. If we tried this with choice A, then we'd get y = 18 which helps eliminate choice A.
Furthermore, plug in x = -2 into choice B and you should get y = 0 as an output. The same applies to x = 3 and x = 1. This confirms that they are roots or x intercepts of the polynomial.
Another way to verify the answer is to use something like WolframAlpha to type in (x+2)(x-3)(x-3)(x-1). Under the "expanded form" subsection, it says x^4 - 5x^3 + x^2 + 21x - 18