How do i do this math problem?

Question

Determine a and b so that when X^4+X^3-7X^2+ax+b is divided by (x-1)(x+2), the remainder is 0?

X^2 means X to the 2nd power.

Please help me, and try showing work if you can.

-thanks.

thanks sniridhi. Its not really hw. its extra credit. I'm still very confused. whats the next step? please? lo, just tell me how to do the entire problem?

So whose answer is correct? ditt or nicewidk?

you guys thanks ALOT for your help. I really appreciate it.

nicewkn: dude how the hell do you post on this without me editing my post? and also, ya i agree he did make a mistake while foiling that out.

And two of the people came up with the same answers. Just their Negative signs are messed up. is it 5 or neg 5?

Okay thanks guys SOOOOOO MUCH!

so i guess the answer is -5 and 10. =)

Answer 1

I can give this a try.....

I am assuming that x-1 is a root of this equation.

(1)^4 + (1)^3 - 7 (1)^2 + a(1) +b = 0

+1 +1 - 7 + a + b = 0

a + b = 5

(-2)^4 + (-2)^3 - 7 (-2)^2 + a (-2) + b = 0

16 - 8 - 28 - 2a + b = 0
-2a + b = 20

So now I have two equations and two unknows

b = 20 + 2a

Put this back into the other equation

a + b = 5

a + 20 + 2a = 5

3a = -15

a = -5
b = 10

Put the values back into the original equation and prove the answer is correct.


Edit: I am reading the different answers myself. Hold on I will post back. I think ditlit approach is also good. But I think he made a mistake when he multiplied the two terms.

(x-1) (x+2) = x^2 + x - 2

Do you agree with me??

Perhaps my answer maybe wrong. I am not sure. However I have read the other answers. What I would suggest that you do is to take the answer and divide the equation by (x-1) and (x+2) and see if it works out. The most important thing to learn is to check your work. Finally I got my answer to match. I don't know how i am doing anything unique all I do is edit and post again. Hope this helps. As I have show there is another way to do this problem.

Answer 2

I really shouldn't do this for you since it's such a nice problem.

We have from the premise that:
(x-1)(x+2)(jx^2 + kx + m) = x^4 + x^3 - 7x^2 + ax + b

Now, simplifying the left side we get,

(x^2 + x - 2)(jx^2 + kx + m) =
jx^4 + jx^3 - 2jx^2 + kx^3 + kx2 - 2kx + mx^2 + mx - 2m
jx^4 + (j+k)x^3 + (-2j + k + m)x^2 + (-k + m)x - 2m

We know that j must be 1 since the x^4 term of the right side is 1, so that means we have:

x^4 + (k+1)x^3 + (k + m - 2)x^2 + (m-k)x - 2m. Now we know that k+1 must equal 1 because the x^3 term on the right is 1 so that means that k = 0 and we now have:

x^4 + x^3 + (m - 2)x^2 + mx - 2m. Now, (m-2) must equal -7 because the x^2 term on the right has a coefficient of -7 so m = -5 and we have:

x^4 + x^3 - 7x^2 -5x + 10. So, a = -5 and b = 10.

P.S. The people using polynomial long division missed that fact that a first year algebra student should be able to solve this problem if he/she is clever enough.

Answer 3

I don't really want to give you the answer, since it's quite obviously your homework. However, I will get you started on your path by telling you what the next step is. Expand the equation so that it is now X^4+X^3-7X^2+ax+b divided by 2x^2+x-2. How'd I get that? Easy: just simply the x-1 and x+2 by reverse factoring

Answer 4

First multiply out (x-1)(x+2) to get x^2+x-2

Then doing long division, divide the polynomial by x^2+x-2. To do this, divide x^2 into x^4 to get x^2. (i.e. x^2 * x^2 = x^4) Then multiply x^2 to the polynomial you're dividing with, i.e. x^2+x-2, to get x^4 + x^3 - 2x^2.
Subtract this (like you would when doing division with regular numbers) from the larger polynomial you're dividing into. You should get -5x^2 + ax + b.

Divide -5x^2 by x^2 to get -5. Multiply -5 back to x^2+x-2 and subtract:

-5x^2 + ax + b - (-5x^2 -5x +10) = (a+5)x + (b-10).

The solution is a=-5, b=10

To test your answer use your result from division which should be x^2-5, i.e. (x-1)(x+2) divides into (x^4+x^3-7x^2+ax+b) (x^2-5)times. Therefore,multiply out (x-1)(x+2)(x^2-5) to get x^4+x^3-7x^2-5x-10. You can easily see how a=-5 and b=10.

Answer 5

(x-1)(x+2) = x^2-x-2

Long division
x^2+2x+7
-----------------------------
x^2-x-2 / X^4+X^3-7X^2+ax+b
x^4 -x^3-2x^2
-------------------
2x^3-5x^2-11x
2x^3-2x^2-4x
------------------
7x^2 - 7x-14
7x^2-7x-14

So a is -11 and b = -14