Question

Pre-cal 12 Question!
Polynomial equation.



Determine an equation of a polynomial of degree 4 that has -1/2 as a root of multiplicity 3, and 2x^2-x-1as a factor.

Answer 1

p(x)=a(x+1/2)^3 (2x^2-x-1)
p(x)= a (x+1/2)^3 (x-1)(2x+1)
How is it possible as a degree of 4? This would be degree of 5.

Answer 2

Try to examine the (2x+1) factor.
What root does that lead to? :)

Answer 3

x= -1/2

Answer 4

Good good good.
Which means you only need `two more` to get multiplicity 3, ya?

Answer 5

why multiplicity of 3?

Answer 6

Because the question is insisting that it have exactly multiplicity 3 at that root, yes? :o

Answer 7

right :) sorry

Answer 8

so

(x+1/2)^3

Answer 9

You had originally thought of something like this:\[\large\rm p(x)= a\left(x+\frac{1}{2}\right)^3 (x-1)\left(2x+1\right)\]Let's try something interesting.
Let's factor a 2 out of each term in the last bracket.\[\large\rm p(x)= 2a\left(x+\frac{1}{2}\right)^3 (x-1)\left(x+\frac{1}{2}\right)\]
You are correct that this is a quintic equation, which is too much!
So it turns out we have too many (x+1/2) powers, yes?
Because the second degree polynomial that they gave us contained `one of those roots`.

Answer 10

Ignore the 2 I pulled out, just absorb it into the `a` out front.\[\large\rm p(x)= a\left(x+\frac{1}{2}\right)^{\color{red}{3}} (x-1)\left(x+\frac{1}{2}\right)\]Do you see how you can fix your polynomial?

Answer 11

I believe it is p(x)=a(x+1/2)^2 (x−1) (x+1/2)

Answer 12

Ah that looks better! \c:/

Answer 13

but how to take out that extra degree? @zepdrix

Answer 14

yes. try x = -1/2 for 2x^2 - x -1. you will get zero again. that means (2x -1) also a factor of 2x^2 - x -1

Answer 15

@lochana yes you are right

Answer 16

I think you're misunderstanding the question :)
It's not saying `root -1/2 multiplicity 3` AND SEPARATELY `2x^2-x-1`

No no no ^ it's not saying that.
There is overlap in this information.

It's saying `root -1/2 multiplicity 3` some of which might come from this `2x^2-x-1`

Answer 17

oh I see what you mean. I have to adjust the degree of the factors to achieve the polynomial degree of 4. I get it know. :) Thank you!

Answer 18

ya you didn't have to do anything fancy to fix it, just use your eraser ^^ hehe

Answer 19

lol