The division algorithm for polynomials states that for any two polynomials p(x) and g(x), where g(x) ≠ 0, we can find polynomials q(x) and r(x) such that:
p(x) = g(x) * q(x) + r(x)
where:
- p(x) is the dividend (the polynomial being divided).
- g(x) is the divisor (the polynomial by which we are dividing).
- q(x) is the quotient (the result of the division).
- r(x) is the remainder (the polynomial left over after the division).
The division algorithm also specifies that either r(x) = 0 or the degree of r(x) is less than the degree of g(x). In simpler terms, you keep dividing until the degree of the remaining polynomial is less than the degree of the divisor, or until you have a remainder of zero.
Example:
Let's say p(x) = x² + 3x + 5 and g(x) = x + 1. We want to find q(x) and r(x) such that x² + 3x + 5 = (x + 1) * q(x) + r(x).
Performing polynomial long division:
x + 2 (Quotient)
x + 1 | x² + 3x + 5 (Dividend)
- (x² + x)
---------
2x + 5
- (2x + 2)
---------
3 (Remainder)So, q(x) = x + 2 and r(x) = 3. We can verify:
(x + 1)(x + 2) + 3 = x² + 2x + x + 2 + 3 = x² + 3x + 5
This matches our original p(x).
Key Points:
- The degree of the remainder must be less than the degree of the divisor.
- If the remainder is zero, then the divisor is a factor of the dividend.
The division algorithm for polynomials provides a systematic way to divide one polynomial by another, similar to long division with numbers. It's a fundamental concept for understanding polynomial relationships and factorization.
