(a) On the same coordinate axes, sketch graphs (as accu-...

(a) On the same coordinate axes, sketch graphs (as accu- rately as possible) of the functions $y=x^{3}-2 x^{2}-x+2 \quad$ and $\quad y=-x^{2}+5 x+2$ (b) Based on your sketch in part (a), at how many points do the two graphs appear to intersect? (c) Find the coordinates of all intersection points.

(a) On the same coordinate axes, sketch graphs (as accu-
rately as possible) of the functions
$y=x^{3}-2 x^{2}-x+2 \quad$ and $\quad y=-x^{2}+5 x+2$
(b) Based on your sketch in part (a), at how many points do
the two graphs appear to intersect?
(c) Find the coordinates of all intersection points.

Key Concepts

Graphing Polynomial Functions

This concept involves plotting polynomial equations on the coordinate plane to visually understand their behavior, including end behavior, turning points, and intercepts. It emphasizes the importance of recognizing the degree and leading coefficient of a polynomial, which dictate how the graph will rise or fall at the ends and how many turning points are possible.

Intersection Points of Graphs

Finding the points where two graphs meet requires setting the functions equal to each other and solving the resulting equation. These intersection points are the solutions to the system of equations defined by the equals sign, providing both the x- and y-coordinates where the graphs cross.

Solving Polynomial Equations

This involves methods to find the roots of a polynomial, such as factoring, applying the Rational Root Theorem, or using synthetic division and the quadratic formula for lower-degree factors. Mastery of this concept enables one to solve higher-order equations by breaking them down into simpler components.

Factorization Techniques

Factorization is a method to simplify polynomial expressions by rewriting them as a product of lower-degree polynomials. It is a key tool when solving equations since it can transform complex equations into more manageable ones, often allowing the direct identification of solutions through setting each factor equal to zero.

Transcript

00:03 All right, we wish to find the points of intersection for the two graphs of y equals x cubed minus 2x squared minus x plus 2 and y equals negative x squared plus 5x plus 2.

00:32 So we would want to sketch these two graphs and there's a couple of different ways we could do this.

00:38 We could use what we know about polynomials, about zeros and behavior, roots, and all of that.

00:48 And we could find those points of intersection.

00:50 I'm going to use, with the help of d 'esmos a little bit, i'm going to graph, graph this, and find the points of intersection.

00:58 So we're going to get an idea of what these graphs look like.

01:03 So i'm going to type in y equals x cubed.

01:19 Minus 2x squared minus x plus 2.

01:27 And we get that cubic polynomial right there.

01:32 And then i'm also going to type in negative x squared plus 5x plus 2.

01:45 And we get that right there.

01:50 So right now we want to find the points of intersection.

01:54 It looks as though there are two right now, but we have to be cautious with our viewing window if we're going to use a graphing utility to do this, because what we're going to find, if we zoom on enough, that there's actually a third point of intersection that we have to be concerned with.

02:12 So our curve, we have one curve that is doing something kind of like that, and we have another curve that's doing something...

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