Division algorithm for polynomials

In summary, the division algorithm for integers states that for positive integers M and N, there exists integers Q and R such that M=QN+R with 0≤R<N. The division algorithm for real polynomials states that for real polynomials xM - 1 and xN - 1, there exist real polynomials q and r such that xM - 1 = q(xN - 1) + r with r = 0 or deg r < N. To find q and r, the division algorithm can be used by dividing xM - 1 by xN - 1, resulting in q(x) = 1 + x + x^2 + ... + xm-1 and r =
  • #1
Kate2010
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Homework Statement



M and N are positive integers with M>N. The division algorithm for integers tells us there exists integers Q and R such that M=QN+R with 0[tex]\leq[/tex]R<N. The division algorithm for real polynomials tells us that there exist real polynomials q and r such that xM - 1 = q(xN - 1) + r with r = 0 or deg r < N. Find q and r.

Homework Equations





The Attempt at a Solution



I have rewritten it as xQN+R - 1 = q(xN - 1) + r, so I think q must be of degree QN+R-N = N(Q-1)+R

However, it is not then always (in fact more likely not) true that N(Q-1)+R<N so 1 must have more than 1 term. But I am struggling to know where to go from here.

Thanks for any help :)
 
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  • #2
Take p(x) = xm - 1. What are the roots of p(x)? There is a clearly a root, 1, so x-1 is a factor. Now divide p(x) by x-1.
 
  • #3
So I get to

xM-1 + xM-2 + ... + 1 = q(xM-1 + xM-2 + ... + 1) + r

But I don't know what to do now?
 
  • #4
You are trying to figure out q and r in: xM - 1 = q(xN - 1) + r

You are aware that x-1 is a factor, which is clearly of the form xN - 1 (N=1). That gives you a hint on what the degree of r is.
You have also correctly identified the polynomial q(x) = 1 + x + x2 + .. + xm-1. So what is (x-1)q(x)? Does this give you your original polynomial xM - 1? So then what is r?
 
  • #5
Why don't you use the division algorithm? Practice on something like x^14-1=q(x^3-1)+r. It's actually pretty easy. You just have to replace the numbers with M, Q, R and N.
 
  • #6
Sorted! Thanks for all the help :)
 

What is the division algorithm for polynomials?

The division algorithm for polynomials is a mathematical process used to divide one polynomial by another. It allows us to find the quotient and remainder when dividing polynomials, similar to long division for numbers.

How does the division algorithm for polynomials work?

The division algorithm for polynomials involves dividing the terms of the dividend polynomial by the terms of the divisor polynomial, one at a time. The resulting quotient and remainder are then used to form a new polynomial, which is usually simpler than the original.

What are the steps involved in the division algorithm for polynomials?

The steps involved in the division algorithm for polynomials are:
1. Arrange the polynomials in descending order of degree.
2. Divide the first term of the dividend by the first term of the divisor to get the first term of the quotient.
3. Multiply the first term of the divisor by the first term of the quotient, and subtract it from the dividend. This will give the first remainder term.
4. Bring down the next term of the dividend and repeat the process until all terms have been divided.
5. The final quotient and remainder form the solution.

When is the division algorithm for polynomials used?

The division algorithm for polynomials is used when we need to divide a polynomial by another polynomial. It is commonly used in algebra and calculus to simplify expressions, solve equations, and find roots of polynomials.

What are some common mistakes to avoid when using the division algorithm for polynomials?

Some common mistakes to avoid when using the division algorithm for polynomials include:
- Forgetting to arrange the polynomials in descending order of degree.
- Making a mistake while dividing the terms of the dividend by the terms of the divisor.
- Not properly subtracting the product of the divisor and quotient from the dividend to get the remainder.
- Making a mistake while bringing down the next term of the dividend.
It is important to double check each step and be careful with calculations to avoid these mistakes.

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