The remainder in polynomial division is written as the term left over after dividing one polynomial, p(x), by another, d(x). You express the result of polynomial division in the form of a quotient q(x) and a remainder r(x), where p(x) = q(x) d(x) + r(x)*.
In simpler terms, you express the original polynomial p(x) as the product of the divisor d(x) and the quotient q(x), plus the remainder r(x).
Here's a breakdown:
- p(x): The polynomial being divided (the dividend).
- d(x): The polynomial you are dividing by (the divisor).
- q(x): The result of the division (the quotient).
- r(x): The polynomial left over after the division (the remainder).
The Equation:
According to the provided reference, the relationship between these polynomials is expressed as:
- p(x) = q(x) d(x) + r(x)*
How to Express the Remainder:
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Perform Polynomial Long Division: Use the long division method to divide p(x) by d(x).
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Identify the Remainder: The remainder, r(x), is the polynomial that's left over at the end of the long division process. The degree of r(x) must be less than the degree of d(x).
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Write the Result: Express the result of the division using the equation p(x) = q(x) d(x) + r(x). Alternatively, you can rearrange the equation to isolate the remainder conceptually, representing it as the part of p(x)/d(x) that is not q(x)*.
Example:
Let's say we divide p(x) = x2 + 3x + 5 by d(x) = x + 1. Through polynomial long division (not shown here for brevity), we find:
- q(x) = x + 2
- r(x) = 3
Therefore, we can write:
- x2 + 3x + 5 = (x + 2)(x + 1) + 3
In this case, the remainder is 3.
