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which graph represents the polynomial function f(x) = x^3 + 3x^2 - 10x -24?

First graph [top left]

Step-by-step explanation:

The first step to figuring this is to FACTOR this polynomial expression. By the Rational Root Theorem, looking at −24, we can tell that −2 [from the expression of (x - 2)] is a root to this expression, so we set it up like this, then use what is called Synthetic Division, since the divisor is in the form of x - c:

Once again, since the divisor is in the form of x - c, use what is called Synthetic Division. Remember, in this formula, -c gives you the OPPOSITE terms of what they really are, so do not forget it. Anyway, here is how it is done:

−2| 1 3 −10 −24

↓ −2 −2 24

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1 1 −12 0 → x² + x - 12 >> [x - 3][x + 4]

You start by placing the c in the top left corner, then list all the coefficients of your dividend [x³ + 3x² - 10x - 24]. You bring down the original term closest to c then begin your multiplication. Now depending on what symbol your result is, tells you whether the next step is to subtract or add, then you continue this process starting with multiplication all the way up until you reach the end. Now, when the last term is 0, that means you have no remainder. Finally, your quotient is one degree less than your dividend, so that 1 in your quotient can be an x², another 1 [x] followling right behind it, and finally comes −12, giving you the other factor(s) of x² + x - 12. This is factorable, so you have to go even further and factor this:

Do not forget that you have one more factor to this from the start, especially since because our degree is 3 [x + 2]:

Now that we have our factored expression, we have to look for two things: the graph has a y-intercept of −24 and that the graph has three x-intercepts of 3, −4, and −2 (from when you set your factored expression equal to zero). With all this being said, you have your answer.

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