Given a polynomial function, explain how to determine whether an x -intercept is a touch point o

Given a polynomial function, explain how to determine...

Key Concepts

Behavior at X-intercepts (Touch vs Cross Points)

The behavior of a polynomial at an x-intercept depends on the multiplicity of the corresponding root. When the multiplicity is even, the graph touches the x-axis at the intercept and leaves on the same side, forming what is known as a touch point. Conversely, when the multiplicity is odd, the graph crosses the x-axis, forming a cross point.

Multiplicity of a Root

The multiplicity of a root refers to the number of times a particular solution appears as a factor in the polynomial. It determines the nature of the x-intercept; if the multiplicity is one (or any odd number), the graph typically crosses the x-axis, and if it is two (or any even number), the graph usually touches the x-axis and turns around.

Polynomial Function

A polynomial function is an expression made up of variables and coefficients that involve only the operations of addition, subtraction, multiplication, and non-negative integer exponents. It is fundamental in algebra and calculus as it exhibits continuous behavior and has a well-understood structure based on its degree and the multiplicity of its roots.

X-intercept

An x-intercept is a point where the graph of the function crosses or touches the x-axis, meaning the function value is zero at that point. Identifying x-intercepts is crucial in sketching the graph of a function and understanding its behavior in relation to the coordinate axes.

Transcript

00:03 And this question asks us, how do we tell if an x intercept is a touch point or a cross point? what this means is we've got a touch point.

00:15 Let's just say negative 3 is a 0 and 2 is a 0.

00:19 If it's a cross point, it's going to go through that intercept and turn and come back down at some point.

00:26 That would be a 0 as well.

00:28 So through points, that's a cross point.

00:31 If it goes through the x intercept.

00:36 A touch point instead of going through, it's going to, i call it a bounce point, but it's going to bounce off of that point.

00:44 So instead of going through, it's going to bounce off, and then turn, and then maybe go back and through.

00:50 Or this one could go, it could bounce off here, and then it could bounce off here.

00:56 And in that case, both of those would be touch points and not cross points.

01:00 The way you tell is you want to look at something called the multiplicity of the factor or the root.

01:08 So if we have a function f of x and we factor it and it factors to be x minus 1 times x plus 3, then 1 is a root and negative 3 is a root and it's only a root once.

01:24 This has a multiplicity of 1 and this has a multiplicity of 1 because it's only a root once.

01:31 So those would be cross points.

01:34 However, if i made this x plus 3 squared, then x is going to be a root, or negative 3 is going to be a root twice.

01:42 So now it has a multiplicity of 2.

01:44 And when it's an even number multiplicity, that's a touch point.

01:51 So basically, an odd multiplicity is a cross point, an even multiplicity is a touch point...

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