Which statement best describes the graph of x³ - 3x² - x + 3?
It starts down on the left and goes up on the right and intersects the x-axis at x = -1, 2, and 3.
It starts down on the left and goes up on the right and intersects the x-axis at x = -1, 1, and 3.
It starts up on the left and goes down on the right and intersects the x-axis at x = -1, 2, and 3.
It starts up on the left and goes down on the right and intersects the x-axis at x = -1, 1, and 3.

Solution:

Given expression x³ - 3x² - x + 3

Let y = x³ - 3x² - x + 3

In order to find the nature of the graph, we need to plot the graph

To get graph, take sample points for (x, y) and plot them on X-Y axis.

In order to know where the graph intersects x-axis, we need to put y = 0

y= x³ - 3x² - x + 3 = 0

The sum of all coefficients of given cubic equation: x³ - 3x² - x + 3 = 0 is zero

Hence x = 1 is a root of given cubic equation i.e. (x - 1) is a factor of x³ - 3x² - x + 3

Now, cubic algebraic polynomial:

x³ - 3x² - x + 3 can be factorized as follows

= x³ - 3x² - x + 3

= x²(x - 1) - 2x(x - 1) - 3(x - 1)

= (x - 1)(x² - 2x - 3)

= (x - 1)(x² - 3x + x - 3)

= (x - 1)(x(x - 3) + (x - 3))

= (x - 1)(x - 3)(x + 1)

Hence, the solution of given cubic equation will be given as (x - 1)(x - 3)(x + 1) = 0, x = -1, 1, 3

Therefore, the graph intersects x-axis at x = -1, 1, 3

It starts down on the left and goes up on the right and intersects the x-axis at x = -1, 1, and 3.

Which statement best describes the graph of x³ - 3x² - x + 3?

Summary:

For expression x³ - 3x² - x + 3, the graph starts down on the left and goes up on the right and intersects the x-axis at x = -1, 1, and 3.