Question

y = x² - 2x - 5; y = x³ - 2x² - 5x - 9; 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? enter your answer in the box. y =

Explanation:

Step1: Set the equations equal

To find the intersection point, set \( y = x^2 - 2x - 5 \) equal to \( y = x^3 - 2x^2 - 5x - 9 \):
\[
x^2 - 2x - 5 = x^3 - 2x^2 - 5x - 9
\]

Step2: Rearrange into standard polynomial form

Move all terms to one side:
\[
x^3 - 3x^2 - 3x - 4 = 0
\]
We can try rational roots using the Rational Root Theorem. Possible rational roots are factors of 4 over factors of 1, so \( \pm1, \pm2, \pm4 \).
Testing \( x = 4 \):
\[
4^3 - 3(4)^2 - 3(4) - 4 = 64 - 48 - 12 - 4 = 0
\]
So \( x = 4 \) is a root.

Step3: Factor the polynomial

Since \( x = 4 \) is a root, we can factor \( (x - 4) \) out of \( x^3 - 3x^2 - 3x - 4 \) using polynomial division or synthetic division. Using synthetic division with root 4:
\[

$$\begin{array}{r|rrrr} 4 & 1 & -3 & -3 & -4 \\ & & 4 & 4 & 4 \\ \hline & 1 & 1 & 1 & 0 \\ \end{array}$$

\]
So the polynomial factors as \( (x - 4)(x^2 + x + 1) = 0 \). The quadratic \( x^2 + x + 1 \) has discriminant \( 1 - 4 = -3 < 0 \), so only real root is \( x = 4 \).

Step4: Find the y - coordinate

Substitute \( x = 4 \) into \( y = x^2 - 2x - 5 \):
\[
y = 4^2 - 2(4) - 5 = 16 - 8 - 5 = 3
\]

Answer:

\( 3 \)