Need help with polynomial function.

Hey guys, new here.

So I am having issue's solving this one: **The polynomial 6x^3 + mx^2 + nx - 5 has a factor of x+1. When divided by X-1, the remainder is -4.**

Has anyone solved this one before, or can they now please so I can learn it and move on.

Thanks guys

Re: Need help with polynomial function.

First of all, that is not a problem, because you are not even asking for anything. Are you trying to figure out m and n? Secondly, can we see the work you have already done? The simple method here is synthetic division, which I suppose you have learned since you are coming across this type of question. You can utilize synthetic division and the two pieces of information on x+1 and x-1 to develop a system of equations.

Please show us the work you have done so far on the problem.

Re: Need help with polynomial function.

Sorry, yes m and n is what I am looking for. I do know synthetic division, but I am not sure even what to divide in this problem. That is my issue. Can you push me onto the right path and I will update when I have moved along?

Re: Need help with polynomial function.

Am I going to be diving the polynomial by x+1?

Re: Need help with polynomial function.

So you know that since x+1 is a factor, the remainder is 0. Work backwards from 0 and forwards from 6 to make one system of equations. The next system of equations you can make by just do normal synthetic division with x-1, and you know that the remainder you get that has m and n in it is equivalent to -4.

Re: Need help with polynomial function.

What values am I changing when you say "backwards form 0 and forwards from 6"? Could you provide an example please.

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Re: Need help with polynomial function.

I'll show you the synthetic division of x+1

You know that at the end, it must be 5 to make the remainder zero, and the sum before must be -5 since reversing the multiplication would be dividing by -1, making -5. When you work from the beginning, at the same sum you get 6-m+n. So that means

6-m+n = -5. You do something similar for x-1.

Re: Need help with polynomial function.

Quote:

Originally Posted by

**mathfish101** So I am having issue's solving this one: **The polynomial 6x^3 + mx^2 + nx - 5 has a factor of x+1. When divided by X-1, the remainder is -4.**

Has anyone solved this one before, or can they now please so I can learn it and move on.

You are expected to use both *the factor theorem* and *the remainder theorem*.

Let $\displaystyle f(x)=6x^3+mx^2+nx-5$. From the given you know that $\displaystyle f(-1)=0~\&~f(1)=-4~.$

Now solve for $\displaystyle m~\&~n$.

Re: Need help with polynomial function.

It isn't **necessary** to use "synthetic" division, that's just a quick way of doing simple divisions. The fact that "x+ 1 is a divisor" means that if you divide $\displaystyle 6x^3+ mx^2+nx- 5$ by x+ 1, the remainder is 0. So go ahead, divide by x+ 1 and see what the remainder is (it will depend on m and n). Set that equal to 0.

The fact that "divided by x- 1 the remainder is -4" means you can divide by x- 1 and see what the remainder is (it will depend on m and n). Set that equal to -4.

You now have two equations to solve for m and n.

Re: Need help with polynomial function.

That is true, my mind is just set that way. It is just as easy to plug in 1 and -1 to the function.

Re: Need help with polynomial function.

Oh got it, I have to plug in actual variables m and n, I kept just plugging in 1. I was wondering why nothing was making sense.

Thanks for the help guys. :)

Re: Need help with polynomial function.

remember the remainder theorem. It states that when a polynomial f(x) is divided by a linear polynomial (x + a ) then the remainder is given by f(-a). also remember that if ( x + a ) is a factor of f(x) then when we divide the polynomial by ( x + a ) the remainder will be zero i.e., f(-a) = 0

Now we have ( x+ 1 ) as factor of the polynomial f(x) = 6x^3 + m x^2 + nx - 5 Thus f(-1) = 0

That gives -6 + m - n - 5 = 0 OR m - n = -11

In second case it is given that when polynomial f(x) is divided by ( x - 1 ) the remainder is -4. That gives us f(1)= -4

That is 6 + m + n - 5 = -4

OR m+n = -5 Now we can calculate m and n from the two equations that we have got.