Question

Help me with this Algebra 2 Honors Assignment??
Graphing involved.

3. 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.
4. Without plotting any points other than intercepts, draw a possible graph of the following polynomial:
f(x) = (x + 8)^3(x + 6)^2(x + 2)(x – 1)^3(x – 3)^4(x – 6).

Answer 1

the question is not really clear
are there exponents there? if they are all exponents there are way too many of them to be a seventh degree polynomial

Answer 2

@satellite73 I fixed it.

Answer 3

@Kawaiicat123 Please help me. I have to get this done TODAY. ;A;

Answer 4

heres your graph for # 4

Answer 5

Those are exponents(I fixed the question), not multiplication.. Could you please update that one..? :D

Answer 6

@Mathmale Can you please help me?? :(

Answer 7

Hello, Hedgie,
Glad to help.
Take a good, hard look at the graph. If the graph CROSSES the x-axis at a certain x-value, the multiplicity of that root is likely to be 1. On the other hand, if the graph touches but does not cross the x-axis at a given x-value, the multiplicity of that root is likely to be 2.
Categorize the roots you see in this illustration.
The roots that have possible multiplicity 1 are ... ??
The roots that have pos. mult. 2 are ... ??

Answer 8

Hedgie, are you absolutely positive that we're talking about a 7th order poly? Wouldn't a 7th order poly have 7 roots (whether real or complex), not just 6, and wouldn't the highest power be x^7?

Answer 9

Okay, um, I think the ones with Mult. 1 are.. (4, 0) and (7, 0)
And the one with mult 2 is (-1, 0).
I'm probably wrong, though, aren't I?

Answer 10

Oh, no wait, hang on.
Mult. 1 are (4, 0), (7, 0) and (-1, 0) while mult 2 is (-5, 0).. Right??

Answer 11

Hedgie, why on earth would you automaticallyl assume that you're "probably wrong"? I'd bet you're right many times more often than you are wrong. Your only sin is one of omission: I believe that the root at -5 also has multiplicity 2.

Answer 12

So I see you got that on your own. Why change y our mind about the multiplicity of the root x=-1?

Answer 13

Let's go back and review this systematically.
Looking at the graph, at which x-values does the graph touch the x-axis but not cross it?
At which does the graph cross the x-axis?

Answer 14

I automatically assume I'm probably wrong because when it comes to math, I usually am. .-. I work off logic and common sense, not numbers. And I changed my mind because it crosses the x axis, and doesn't just touch it.. But by that thought, I guess (4, 0) just touches it as well, because it follows the axis for a point and then goes back off.

Answer 15

You wouldn't be in Alg. II Honors were you a dummy. You're not a dummy. Please let's go back and "review this systematically," as I've suggested above.

Answer 16

I'm in Honors because that's where my school put me, because of how bad my grades in math have been. That's what my mom tells me, anyway. I personally think it was just a mistake.
But okay, let's go back.
Ones that cross the x axis: (-1, 0), (7, 0).
Ones that touch the x axis: (-5, 0), (4, 0).
Right??

Answer 17

I find myself wondering about that x=4 and unsure of the multiplicity there.
For Part I: Why don't we identify the four roots as you have (above), and accept only tentatively the multiplicities that stem from whether or not the graph crosses the x-axis at a given point?

I'd like to move on to Part 2; we could return to Part 1.
Willing to do that?

Answer 18

Sure, we can do that! As long as we get this done, I don't really care. ^^;;;

Answer 19

If you're familiar with the Equation Editor (below), please type the polynomial in Part 2 in it. I find some ambiguities in f(x) = (x + 8)^3(x + 6)^2(x + 2)(x – 1)^3(x – 3)^4(x – 6).

Answer 20

Okay, I can do that. :)
\[f(x) = (x + 8)^3(x + 6)^2(x+2)(x - 1)^3(x-3)^4(x-6)\]

Answer 21

So much clearer this way! Look carefully. What order do YOU think this poly has, and why?

Answer 22

BRB

Answer 23

If I'm supposed to re-arrange them, by exponent.. It'd be to
\[f(x) = (x+2)(x-6)(x + 6)^2(x + 8)^3(x-3)^4\]
by order of smallest to largest.. As for why, really because that's what I think I should do, and I kinda find it funny when numbers in equations look to be like they're counting(as in then they're just one higher or lower from each other, in succession, like (x + 1)(x + 2)(x + 3) and so on.).. Because I'm just silly like that.

Answer 24

Quick review: which order does the folloing poly have? x
Which order: 3x^3-1
Which order? -2x5+x^3 -29

Answer 25

Which order: (x-2)(x-3)(x+5)

Answer 26

I'm.. Not sure, honestly.

Answer 27

Sorry for the slow response; I didn't see your response.
We need to review "order" as it applies here. Look at the four polys I've given you. The highest power in the first is 1, so we dub it a 1st order poly.
The highest power in the 4th is 2, so we dub it a 2nd order poly.
What about the 3rd poly? Of what order is it?
Good idea to double check your textbook or look up "order of a polynomial" online.

Answer 28

For example: http://en.wikipedia.org/wiki/Degree_of_a_polynomial

But anyway, check into that if you have the time and interest.

Let's go back to Part 2.

Answer 29

It's alright, no worries.
Could you please put the polys again? I don't see them. .-.

Answer 30

f(x) = (x + 8)^3(x + 6)^2(x + 2)(x – 1)^3(x – 3)^4(x – 6).

Please look at the first factor: (x+8)^3. the degree here is 3.
The 2nd factor : (x+6)^2. degree is 2.
Continue through the whole thing, adding together the number of degrees pertaining to each factor. I get 15. But you do it yourself, please.

Answer 31

I get 12, for that.. Unless I'm supposed to be counting the invisible ^1's, as well.. Then I get 15, too.

Answer 32

Right. You do have to count EVERY power. So 15 is right. Note, Hedgie, that this is an ODD degree. Remember, x, x^3, x^5, x^7, and so on, are odd powers of x.

If you'd sketch a couple of these really quickly, you'd see that all have graphs that begin in QIII and end in Q1. Are you familiar with the terminology I'm using here?

Answer 33

You don't have to show me your graphs; I just need for you to understand the principle involved.

Answer 34

I'm assuming you mean the quadrants of a graph. Right?

Answer 35

Let me respond with a graph showing the four quadrants.

Answer 36

Okay.