Find y coordinate of intersection

Question

I am having trouble because one of them is a cubic polynomial.

y= x^2 - 2x - 5
y= x^3 - 2X^2 - 5x - 9

Question
When the two equations shown are graphed in the coordinate plane, they intersect at a point. What is the y -coordinate of the point of intersection?

Answer 1

Set the polynomial equal to each other.

 

x² - 2x - 5 = x³ - 2x² - 5x - 9

 

 

Move all the terms to the right side of the equation to make the left side equal to zero.

 

0 = x³ - 3x² - 3x - 4

 

 

Now we use synthetic division to factor.

 

 4 |  1       -3        -3           -4

 

                 4         4             4

___________________________

        1       1          1            0

 

 

0 = (x - 4)(x² + x + 1)

 

 

Now you can set the factors equal to zero and solve for x.   Since the quadratic factor has no real solutions, we ignore and focus on the linear factor.

 

x = 4

 

Evaluate any of the function when x=4 to get the y-coordinate of the intersection point.

Answer 2

Hi Tia,

 

The intersection point will be where the two equations will yield the same value, so we can write

x^3 - 2x^2 - 5x - 9 = x^2 - 2x - 5

x^3 - 3x^2 - 3x -4 = 0 

 

I tried but there doesn't seem to be any way to factorize this polynomial further. 

I will try synthetic division with a few numbers (maybe up to 5) to see if any of them is a root of the polynomial. Remember, according to the Remainder Theorem, you should get a remainder of zero if the divisor is a prefect factor of the polynomial. 

You will need to refresh your memory on synthetic division. When you try, you will find out that 4 is a factor bacuse the remainder is zero. 

 

4| 1   -3   -3   -4

  |____4__4___4________

    1    1    1     0

 

The quotient is x^2 + x + 1. We can see that this second degree polynomial has no real roots because the discriminant is less than zero. b^2 - 4ac = 1^2 - 4*1*1 = -3

The only real number x value is 4.

Then, when we plug this in one of the equations given in the question, we can get the y value of the intersection.

y= x^2 - 2x - 5

= 4^2 - 2*4 - 5

= 3

 

This is my approach to solve this problem. There may be more than one solution. I would say that this is a rather involved Algebra 1 question since it requires the integration of multiple concepts for the solution. 

Answer 3

Graph them both (easy to do if you have a graphing calculator and can put in both equations; otherwise, find an online graphing calculator like desmos).

 

The point of intersection is (4,3).  

 

You could also make a short table of values for each, and then see when the same x-coordinate gives the same y-coordinate for both equations.