Degree of Remainder When Dividing a Fourth Degree Polynomial by a Quadratic Polynomial

When considering polynomial division, the degree of the remainder is a critical aspect to understand. Specifically, when a polynomial of degree ( n ) is divided by another polynomial of degree ( m ), the degree of the remainder is at most ( m - 1 ). This principle is fundamental in polynomial arithmetic, particularly in the context of dividing higher-degree polynomials by lower-degree ones.

Understanding Polynomial Division

Let’s consider the polynomial ( P(x) ) of degree ( n 4 ). When ( P(x) ) is divided by a quadratic polynomial ( D(x) ) of degree ( m 2 ), the remainder ( R(x) ) must have a degree less than ( m ).

Mathematically, the division can be expressed as:

P(x) Q(x) cdot D(x) R(x)

where ( Q(x) ) is the quotient polynomial, and ( R(x) ) is the remainder. The degree of ( R(x) ) is at most ( m - 1 2 - 1 1 ). Hence, ( R(x) ) can be a polynomial of degree 0 (a constant) or degree 1 (a linear polynomial).

Example and Explanation

Let’s illustrate this with an example. Consider ( P(x) x^4 2x^3 - x 3 ) and ( D(x) x^2 - 1 ).

When we divide ( x^4 2x^3 - x 3 ) by ( x^2 - 1 ), the quotient and the remainder are determined as follows:

x^4 2x^3 - x 3 (x^2 2x 2) cdot (x^2 - 1) (x 1)

In this case, the remainder ( R(x) x 1 ) is a linear polynomial, confirming that the degree of the remainder is no more than 1.

General Principle and Proof

The principle that the degree of the remainder is at most ( m - 1 ) can be derived from the well-established polynomial division algorithm. When performing the division, the degree of the remainder is reduced by each step of the division process. This ensures that once the degree of the divisor is reached, the division process can no longer proceed, and the remainder must have a degree less than the divisor.

For example, if we divide a fourth-degree polynomial by a second-degree polynomial, the division process will reduce the degree of the remainder until it is less than 2. Therefore, the remainder is either a constant (degree 0) or a linear polynomial (degree 1).

Conclusion

Understanding the degree of the remainder in polynomial division is crucial for various applications in algebra and calculus. The degree of the remainder is at most the degree of the divisor minus one, which simplifies the process of polynomial division and ensures that the division is properly completed.

When dividing a polynomial of degree 4 by a quadratic polynomial, the remainder will always be a polynomial of degree 1 or less, typically a linear or a constant term.