Hey, everyone it’s Shanise from Ms. Bruchat’s period 3/4 class. I hope everyone is enjoying and understanding polynomial functions so far! Here is a review of Friday’s lesson.
An equation in factored form can reveal a lot about the equation:
1. each factor gives us the value of the graph’s zeros(x-intercepts)
Ex. y= (x-3)(x+1)(x-4) Make y=0 and solve for x to find the zeros
Values of zeros are x= 3,-1, 4
2. determines the number of zeros
For example the same equation from above has three brackets or three zeros. This tells us the number of zeros in this equation is three.
Ex. y= (x-3)(x+1)(x-4)
First zero- (x-3) =0
Second zero(x+1) =0
Third zero (x-4) =0
3. helps us to determine the degree of the polynomial
In this example to determine the degree of a polynomial you must count all the x’s.
Ex. y= -4(x+3) (x-2)^2(x+6) (x+2)
We must take the exponent “2” into account on the second bracket, when we are counting. In all there are four x’s, which means that this is a degree 4 polynomial.
4. the multiplicity(order) of each factor tells us how the graph interacts at the x-intercepts
To find the order of each factor, it is the exponent after the brackets and if there is no exponent on bracket/brackets then the order is one. For example this equation y=(x-1)(x+2) has an order of one.
Multiplicity of 1- a straight line travels through the zeros
Multiplicity of 2(4, 6, 8…) - bounces at the zeros
Take this equation for example y=(x-1) ^2 (x+2) at the x-intercept x=1, it will bounce and a straight line will go through the other zero x= -2
Multiplicity of 3(5, 7, 9…) - inflects at the zeros
For this equation y=(x+1) ^3 (x-3), at the x-intercept x=-1 the line going through it will not go straight through but it will be inflected. However for the second zero x=3 the line will go straight through.