What is the meaning of a zero of a polynomial and how is it determined?

[duki]

Homework Statement

Determine the remainder using the remainder theorem
P(x) = 2x^3 + 3x^2 + 4x - 10; D(x) = x + 1

Homework Equations

Remainder Theorem

The Attempt at a Solution

x = -1
P(x) = -2 + 3 - 4 - 10
R -13Can you have a negative remainder?The next question says:
Decide whether or not the number is a zero of the polynomial.

Is that just another way of saying find the remainder?

 

[Dick]

Sure you can have a negative remainder. Is the 'number' you are referring to -1? If so, IS it a root?

 

[duki]

Oh ok, so -13 is the right answer to that one?


No, the next question is referring to a different polynomial, though the question was stated different. Can I use the remainder theorem again, or is it asking me to do something different?

 

[Dick]

What's the remainder of x-r if r is a root? And if you don't believe -13 is the remainder - check it with a long division.

 

[HallsofIvy]

The remainder theorem which you mention but don't cite (that would have been a good idea) says that if P(x) is divided by (x-a) then the remainder is P(a). That's true because P(x)/(x-a)= Q(x)+ remainder r is the same as saying P(x)= (x-a)Q(x)+ r. What happens when you set x= a?

 

[duki]

I'm lost =/

My teacher didn't go over the theorem, he just told us how to use the f(x) = blah

Sorry I didn't site it.

I'll work it out in long division to see if I got the right answer, but I'm thinking I did since you can have a - remainder.

Now I'm not sure what to do about the question that asks if a certain number is a zero of the polynomial... is that the same as saying find the remainder?

 

[Dick]

Look at what Halls said. Dividing a polynomial P(x) by (x-a) and getting a remainder r and a quotient Q(x) means you can write the polynomial P(x)=Q(x)*(x-a)+r, right? Make sense? Just the same as division by numbers? Putting x=a in that gives:

P(a)=r

since a-a=0. That is the Remainder theorem. So to see if r=0 (which is the same as saying that a is a root, yes?), you can either put 'a' into the polynomial and see if you get zero or you can divide the polynomial by (x-a) and see if the remainder is zero. Same thing. Do actually try out the long division to prove it to yourself.

 

[duki]

What is a?

When i divided it with long division I got -9 =/
But when i used synthetic division i got -13 again...

The question about finding a zero is

P(x) = -x^4 + 9x^2 - 9x + 27; 3
So I did used
P(3) = -(3)^4 + 9(3)^2 - 9(3) + 27 = 0
so yes it is a zero?

There is yet another problem that says find the quotient and remainder of the following problems and whatnot... i can get the remainder but how do i get the quotient?
Is that where I subtract a root from the equation since I just took one out?
If so here is what I did:

Problem: (3x^5 + 4x^4 + 2x^2 - 1) / (x + 2)

Solution (using synth div.)

3 -2 4 -6 R11
so
3x^3 - 2x^2 + 4x - 6 R11 ??

 

[HallsofIvy]

1. Homework Statement

Determine the remainder using the remainder theorem
P(x) = 2x^3 + 3x^2 + 4x - 10; D(x) = x + 1

2. Homework Equations

Remainder Theorem

3. The Attempt at a Solution
x = -1
P(x) = -2 + 3 - 4 - 10
R -13

In the statement of the problem you said the the polynomial was 2x³+ 3x²+ 4x- 10. Here you use "-2 + 3 - 4 -10" which corresponds to -2x³+ 3x²- 4x- 10. Which is it?

The question about finding a zero is

P(x) = -x^4 + 9x^2 - 9x + 27; 3
So I did used
P(3) = -(3)^4 + 9(3)^2 - 9(3) + 27 = 0
so yes it is a zero?

? If you were asked to find a zero, where did you get "3" from? If you were asked to determine whether or not 3 is a zero then it's just a matter of understanding what a "zero" of a polynomial is.

There is yet another problem that says find the quotient and remainder of the following problems and whatnot... i can get the remainder but how do i get the quotient?
Is that where I subtract a root from the equation since I just took one out?
If so here is what I did:

Problem: (3x^5 + 4x^4 + 2x^2 - 1) / (x + 2)

Solution (using synth div.)

3 -2 4 -6 R11
so
3x^3 - 2x^2 + 4x - 6 R11 ??

Are you saying you do not know the meaning of the word "quotient"? If you were to divide 1233 by 5 what would be the quotient and what would be the remainder?

 

[duki]

In the statement of the problem you said the the polynomial was 2x3+ 3x2+ 4x- 10. Here you use "-2 + 3 - 4 -10" which corresponds to -2x3+ 3x2- 4x- 10. Which is it?

It's the one stated in the problem, but i did (-1)^3 * 2.

? If you were asked to find a zero, where did you get "3" from? If you were asked to determine whether or not 3 is a zero then it's just a matter of understanding what a "zero" of a polynomial is.

Yes, I needed to determine if 3 was a zero.

Are you saying you do not know the meaning of the word "quotient"? If you were to divide 1233 by 5 what would be the quotient and what would be the remainder?

I actually typed the problem wrong... but it's ok I have it right now.

thanks for the help.