Long division is a step-by-step procedure for dividing one polynomial by another polynomial. It is a systematic method for finding the quotient and remainder when dividing one polynomial expression by another. This technique is applicable in the context of both dividing polynomials and partial fraction decomposition.
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The goal of long division is to find the quotient and remainder when dividing one polynomial by another.
The degree of the divisor polynomial must be less than or equal to the degree of the dividend polynomial for long division to be applicable.
Long division involves a series of steps where the divisor is subtracted from the dividend, and the process is repeated with the remainder.
The quotient is obtained by writing the coefficients of the successive remainders, while the final remainder becomes the actual remainder of the division.
Partial fraction decomposition utilizes long division to break down a rational function into a sum of simpler rational functions.
Review Questions
Explain the purpose and general process of long division when dividing polynomials.
The purpose of long division in the context of dividing polynomials is to find the quotient and remainder when one polynomial is divided by another. The general process involves a systematic series of steps where the divisor is subtracted from the dividend, and the process is repeated with the remainder. The coefficients of the successive remainders form the quotient, and the final remainder is the actual remainder of the division.
Describe how long division is used in the context of partial fraction decomposition.
In the context of partial fraction decomposition, long division is used to break down a rational function into a sum of simpler rational functions. The long division process is applied to the numerator and denominator of the rational function to determine the quotient and remainder. The quotient represents the polynomial part of the partial fraction decomposition, while the remainder is used to determine the partial fraction components.
Analyze the relationship between the degree of the divisor and the applicability of long division when dividing polynomials.
For long division to be applicable when dividing polynomials, the degree of the divisor polynomial must be less than or equal to the degree of the dividend polynomial. This is because the long division process involves a series of subtractions, where the divisor is repeatedly subtracted from the dividend. If the degree of the divisor is greater than the degree of the dividend, the division process would not be possible, as the divisor would be too large to be subtracted from the dividend.