Question

Medal to most helpful in finding the correct answer. The following graph shows a seventh-degree polynomial:
Part 1: List the polynomial’s zeroes with possible
multiplicities.
Part 2: Write a possible factored form of the seventh degree function.

Answer 1

@DariusX @Hero @SolomonZelman @dan815 @jim_thompson5910

Answer 2

@retirEEd

Answer 3

The zeroes are where the function crosses or touches the x-axis.
(We'll deal with multiplicity in a minute, let's find the zeroes first)

So I see that the function touches the x-axis at x=-5.
So -5 is one of our zeroes.

Can you list the others?

Answer 4

The zeros of the function are -5, -1, 3.5, 4, 4.5, 7. Correct? @zepdrix

Answer 5

The functions line was drawn a little too thick I suppose :)
It's crossing at x=4, yes, but not those two points around it,
maybe it just looks like it.

So we have our zeroes:
-5, -1, 4, and 7

Answer 6

If you remember the basic shape of some of your polynomials,
it will really help understand what is going on with multiplicity.

Answer 7

Parabola, a second degree polynomial, is a bowl shape, ya?
Do you see that shape at any of our zeroes?

Answer 8

I see it at -5. @zepdrix

Answer 9

Good!
So at -5, our zero has a multiplicity of 2,
because the shape corresponds to a second degree polynomial.

Answer 10

A linear or first degree polynomial is a straight line,
so now let's try to find the locations where the curve doesn't do anything fancy at the zero, it just cuts right through.

Answer 11

Oh ok. That makes sense. I see a straight line going through -1, 3.5, 4, 4.5, and 7.

Answer 12

-1 and 7 sound correct.
But we have some weird funny business going on at 4.
(again you have to let the 3.5 and 4.5 go, we don't care about those, not zeroes)

Answer 13

are they there to throw me off? So, would -1 and 7 have a multiplicity of 1?

Answer 14

Ok great, yes.

Answer 15

At x=3.5, the blue line is just barely below the x-axis
at x=4.5, the blue line is just barely above it.

It only crosses the x-axis at x=4, not those surrounding points.
I don't think they intended to trick you,
it's the fact that the line was drawn too thick.

Answer 16

I'll draw an example of a third degree polynomial just to remind you.

Answer 17

(Sorry, I'm on my laptop, touchpad lol)

Answer 18

This is the kind of shape we're seeing at x=4, ya?

Answer 19

it's maybe drawn a bit more flattened out in your function, but the same shape, ya?