Intersection Points of Two Polynomials (a) On the same coordinate axes, sketch graphs (as a

step 1 So we've been given two graphs here. I've just gone ahead and put these on a graphical calculato

Intersection Points of Two Polynomials (a) On the same...

Key Concepts

Polynomial Functions

Polynomial functions are expressions involving variables raised to whole number exponents and their coefficients. They are fundamental in understanding algebraic behavior and are characterized by their degree, which determines their end behavior, number of turning points, and possible range of values.

Graphical Representation of Functions

Graphical representation involves sketching the graph of a function to visualize its behavior, including intercepts, turning points, and overall shape. This technique is essential for understanding the qualitative aspects of a function and for estimating interaction points with other graphs.

Intersection Points of Graphs

The intersection points of graphs occur where the function values are equal for the same input, which translates to solving the equation f(x) = g(x). This concept is important for analyzing how different functions relate to each other and for identifying common solutions.

Solving Polynomial Equations

Solving polynomial equations involves finding the roots or zeros of the polynomial, which are the values of the variable that satisfy the equation. Techniques such as factoring, the Rational Root Theorem, or the quadratic formula (for quadratics) are commonly used, and are crucial for precisely determining the intersection points of polynomial graphs.

Transcript

00:01 So we've been given two graphs here.

00:03 I've just gone ahead and put these on a graphical calculator.

00:06 You can sketch them out by factorising, but we've already done quite a lot in this topic already, so i thought i just put this on the graph, just to speed things along the time a bit.

00:17 So we've got, in the red, we've got y equals x cubed minus 2x squared minus x plus 2.

00:22 And in the green, we've got y equals minus x squared plus 5x plus 2.

00:28 Okay, so we've done that, we sketch the two graphs or we put them into our graphical calculators.

00:36 And then it says, on the basis of your sketch in part a, at how many points do the two crafts appear to intersect? so we've got one intersection point down here at, well, we'll be figuring that out in a second.

00:48 And we've got another one up here and we've got another one up here.

00:51 So it seems that we've got three intersection points.

00:54 So when we go into part c, which is find the coordinates, all the use.

00:57 Intersection points, we'll be expecting roughly three intersection points.

01:02 And that does make sense because we've got an x cubed here.

01:05 So when we equate that with an x squared, we should be expecting three intersection points.

01:10 Let's get on with this.

01:11 Oh, that's just my email there.

01:15 Okay.

01:15 So let's start with trying to find the three intersection points.

01:20 So we got the way the way we can find these intersection points, all we have to do is make the equations equal to each other.

01:29 So if we just do that now, we'll get x cubed minus 2x squared minus x plus 2 equals minus x squared plus 5x again plus 2.

01:42 So this plus 2 is just going to disappear.

01:45 And we'll bring the minus x squared and the plus 5x to this side.

01:48 So we'll get x cubed minus 2x squared plus x squared plus x squared.

01:57 Minus 5x minus x and that's going to equal zero because you took everything over to this other side now the two is cancel out and we'll combine terms now so we get x cubed minus x squared minus x squared minus 6x and that's going to equal zero we can factorize out on x here and that'll leave us with x squared minus x minus 6 equals 0 okay, so we already have one value for our x, and that's going to be x equals zero.

02:27 If we were to sub this back into, we can sub this back into any of the equations here to get our y.

02:32 A y is going to equal minus x squared, so it's going to be minus zero.

02:37 So it's going to be two, because anything with zero and it's just going to go to zero.

02:41 So our intersection point here is going to be zero, comma, two.

02:45 So that's one of our intersection points.

02:47 And then we're going to solve x squared minus x minus six.

02:50 And we're going to factorise this to solve it...

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