how to find x and y intercepts for F(x)=2x^3+3x^2-8x+3

Question

I will later have to graph this, but first i need to find xand y intercept and i dont know how to go about that

Answer 1

On the y-axis, x = 0. Thus, we have:

F(0) = 2(0)^3 + 3(0)^2 - 8(0) + 3 = 0 + 0  - 0 + 3 = 3

Thus, (0, 3) is the y-intercept.

On the x-axis, y = 0. Thus, we have:

0 = 2x^3 + 3x^2 - 8x + 3
0 = 2x^3 + 5x^2 - 2x^2 - 3x - 5x + 3
0 = 2x^3 + 5x^2 - 3x - 2x^2 - 5x + 3
0 = (2x^3 + 5x^2 - 3x) - (2x^2 + 5x - 3)
0 = x(2x^2 + 5x - 3) - 1(2x^2 + 5x - 3)
0 = (x - 1)(2x^2 + 5x - 3)
0 = (x - 1)(2x^2 - x + 6x - 3)
0 = (x - 1)((2x^2 - x) + (6x - 3))
0 = (x - 1)(x(2x - 1) + 3(2x - 1))
0 = (x - 1)(x + 3)(2x - 1)
x - 1 = 0 or x + 3 = 0 or 2x - 1 = 0
x = 1 or x = -3 or x = 1/2

Thus, the x-intercepts are (1, 0), (-3, 0) and (1/2, 0).