A function is shown below:
f(x) = x3 + 3x2 - x - 3
Part A: What are the factors of f(x)? Show your work. (3 points)
Part B: What are the zeros of f(x)? Show your work. (2 points)
Part C: What are the steps you would follow to graph f(x)? Describe the end behavior of the graph of f(x). (5 points)
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A) (x + 4)( ) = x^3 + 4x^2 – x – 4
I know the first term must be x^2, since x^3/x = x^2:
(x + 4)(x^2 ) = x^3 + 4x^2 – x – 4
I know the second term must be 0, since 4x ^2 - (4 * x^2) = 4x^2 - 4x^2 = 0:
(x + 4)(x^2 + 0 + ) = x^3 + 4x^2 – x – 4
I know the last term must be -1, since -x - (0 * x) = -x - 0x = -x:
(x + 4)(x^2 - 1)
Now, factor the second part. It's a difference of squares, so:
f(x) = (x + 4)(x + 1)(x - 1)<=====ANSWER
B) Set each factor = to 0:
x + 4 = 0; x + 1 = 0; x - 1 = 0
x = -4. -1. and 1<=====ANSWER
C) Plot the 0's. Solve the equation for numbers that are less than -4, between -4 and -1, between -1 and 1, and greater than 1. I chose -5, -2, 0 and 2:
f(-5) = (-5 + 4)(-5 + 1)(-5 - 1) = (-1)(-4)(-6) = -24
f(-2) = (-2 + 4)(-2 + 1)(-2 - 1) = (2)(-1)(-3) = 6
f(0) = 0^3 + 4(0)^2 - 0 - 4 = -4
f(2) = (2 + 4)(2 + 1)(2 - 1) = 18
The end behavior means which way do the ends of the graph point. On the left side, it points down, so it's negative; on the right side, it points up, so it's positive.