Graphs of Polynomial Functions

Question

8

The polynomial of degree , , has a root of multiplicity  at  and a root of multiplicity  at . The -intercept is .

Find a formula for .

Answer 1

Hi.

So multiplicity means the power of each function.

So if the root is x=2 then the factor would be (x-2)

if the other root is x=-1 then the factor would be (x+1)

(x-2)² would show x=2 with multiplicity of 2

(x+1)¹ would show x=-1 with multiplicity of 1

So the formula to use is y= a(x-c)^m(x-d)^m where c and d are roots m is multiplicity and a is the vertical stretch/shrink factor

y =a (x-2)²(x+1)

Now it is time to use the info y int=-2. That will help us find a.

-2 = a(0-2)²(0+1) ( used the point (0,-2) )

-2= a(4)(1)

-2=4a

-2/4 =a

-1/2=a

y= -1/2(x-2)²(x+1)

OR

P(x) =-1/2(x-2)²(x+1)

Then you can multiply it out to show the highest power of x is 3 so that’s why it is degree 3

-1/2 (x²-2x-2x+4)(x+1)

-1/2(x²-4x+4)(x+1)

(-1/2x²+2x -2)(x+1)

-1/2x³-1/2x²+2x²+2x-2x -2

P(x) = (-1/2)x³ + 3/2x² -2

If you can put it into a graphing app, you will see the graph crosses through at -1 and -2 then just touches at 2. Odd multiplicity it goes through it, even multiplicity, it touches and then goes the opposite way it came from.

I hope this helped you! Have a great day!

Mrs. S.