Find the points of intersection of the line y = x+1 with the polynomial y= 3x^3 -2x^2 - 6x-1

Question

Directions:

The problem needs to be solved analytically - although the calculator may be used to identify roots prior to long division

Answer 1

Set the cubic equal to the binomial and collect terms:

3x³ -2x² - 7x - 2 = 0

Use the theorem on rational roots to discover that -1 and 2 are roots.

Use long division or synthetic division to divide out the two roots to leave you with 3x+1=0 so that x=-1/3

Answer 2

You have 2 equations with 2 unknowns, so use any standard method to solve. Substitution looks the easiest.

In the 2nd equation, replace y with x + 1 to get

3x^3 - 2x^2 - 6x - 1 = x + 1

Rearrange and simplify:

3x^3 - 2x^2 -7x - 2 = 0

Use long division or synthetic division to find the first factor. It will end with either 2 or 1, so try (x ± 1), or (x ± 2)

The remaining 2nd degree polynomial will be factorable with real roots. This means there are 3 points of intersection. When you have found all 3, CHECK by plotting the functions.