Remainder when Dividing Polynomials by x2 - 1
In mathematics, polynomial division is a fundamental concept used to divide one polynomial by another. A critical part of polynomial division involves finding the remainder when a polynomial is divided by a quadratic with specific roots. This article provides a detailed explanation and practical examples of how to find the remainder when a polynomial is divided by x2 - 1.
Understanding the Method
The process of finding the remainder when a polynomial is divided by x2 - 1 can be achieved through polynomial long division or directly using the Remainder Theorem. The polynomial x2 - 1 can be factored into (x - 1)(x 1). Hence, the remainder when a polynomial is divided by x2 - 1 is a linear polynomial of the form ax b.
Step-by-Step Example: x991 รท (x2 - 1)
Step 1: Evaluate at x 1
f(1) 1991 - 1 0 1 2
Step 2: Evaluate at x -1
f(-1) (-1)991 - 1 -1 - 1 0
Step 3: Construct the Remainder
The remainder Rx is a linear polynomial of the form ax b.
tR(1) a(1) b 2 tR(-1) a(-1) b 0Step 4: Set up the Equations
From the above two conditions, we have:
ta b 2 t-a b 0Step 5: Solve the System of Equations
From the second equation, we can express b in terms of a:
b a
Substituting b a into the first equation:
a a 2
Therefore, 2a 2 which gives a 1.
Substituting a 1 back into b a:
b 1
Conclusion
The remainder Rox is:
Rx 1x 1 x - 1
Hence, the remainder when dividing x991 - 1 by x2 - 1 is boxed{x - 1}.
Examples and Equations
Let's consider a simpler example to further illustrate the process. Taking the expression x991 - 1 and dividing by x2 - 1, we can also use polynomial long division or the Remainder Theorem as detailed above.
Another approach using the J programming language involves plugging in specific values. For the formula a^2 - 11a^99x, we get:
[-.0 a^2 - 11a^99x] for a ranging from -10 to 10
This results in zeros for all values of x except when x 1, where the result is 2.
Conclusion
In conclusion, the remainder when dividing x2n 1 - 1 by x^2 - 1 is always x - 1, showing a clear pattern. Similarly, for any polynomial in the form x2n - 1 - 1, the remainder when divided by x^2 - 1 is also x - 1.