+1252 a) We must let \(f(x)=0\).
\(3x^3+10x^2-13x-20=0\), and solving we get \(x=-4, -1, \frac{5}{3}\) .
Therefore, the x-intercepts are \(x=-4, -1, \frac{5}{3}\).
b) Let \(x=0\) , and solving we get \(-20\) .
Therefore, the y-intercept is \(y=-20\).
c) This is simple, make a point between \(x=-4\) and \(x=-1\), for example, \((-3, 0)\).
Same for \(x=-1\) and \(x=5/3\) , for example, \((1, 0)\) .
d) As \(x\) goes toward negative infinity, we know that \(x^3\) will always decrease, and vice versa.
e) Your graph may look something like this:
You are very welcome!
:P
+129899
Let's see how CS might have determined the x intercepts (roots)
3x^3 + 10x^2 -13x - 20
By the Rational Roots Theorem ......one possible root is -1
So
3(-1)^3 + 10(-1)^2 - 13(-1) - 20 = -3 + 10 + 13 - 20 = 0
Using synthetic division, we can find the remaining polynomial
-1 [ 3 10 - 13 -20 ]
-3 - 7 20
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3 7 -20 0
So....the remaining polynomial is 3x^2 + 7x - 20
Factoring, we have
(3x - 5) ( x + 4)
Setting each factor to 0 and solving for x we have that the other two roots are
x= 5/3 and x = -4
Then the x intercepts are x = -4, -1 and 5/3
The rest of the answer is as CS has shown....!!!
