SOLUTION: Give the x-intercepts, the table of signs and the sketch of the graph. f(x) = 2x^3

SOLUTION: Give the x-intercepts, the table of signs and the sketch of the graph. f(x) = 2x^3 - 5x^2 + 4x -1

Question 987565: Give the x-intercepts, the table of signs and the sketch of the graph.
f(x) = 2x^3 - 5x^2 + 4x -1
Found 2 solutions by ikleyn, solver91311:
Answer by ikleyn(52847)   (Show Source): You can put this solution on YOUR website!
.
The roots of the polynomial f(x) = are 1 (of the multiplicity 2) and .

The polynomial is factored as = . You can check it directly.

The plot of the polynomial is shown in Figures 1 and 2.

How to get these results? You can guess that 1 is the root.

After this is done, you can make long division by dividing the given polynomial by the binomial (x-1).
In this way you will be able to decrease the degree of the polynomial from 3 to 2.
Then you can find the roots of the obtained quadratic polynomial.

                      Figure 1

                      Figure 2

Could you make the table of signs yourself?

Notice that since the polynomial contains the factor (x-1) in degree 2, the original polynomial does not change the sign at the vicinity of the point x=1.

Answer by solver91311(24713)   (Show Source): You can put this solution on YOUR website!

Per the rational roots theorem, if this function has any rational zeros they must belong to the set

1 | 2 -5 4 -1 2 -3 1 --------------------- 2 -3 1 0

Hence, 1 is a zero, and and are factors.

which factors to

That gives us zeros for the original function at 1, 1, and

-∞ | 1/2 | 1 | ∞ ------------------------------------------------- x - 1 - - + x - 1 - - + 2x - 1 - + + ------------------------------------------------- (2x-1)(x-1)^2 - + +

.
John

My calculator said it, I believe it, that settles it

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