In Exercises 43–46, perform each long division and write the partial fraction decomposition of the remainder term. (x^5+2)/(x^2-1)
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Step 1: Set up the long division by writing as the dividend and as the divisor.
Step 2: Divide the leading term of the dividend by the leading term of the divisor to get the first term of the quotient, .
Step 3: Multiply the entire divisor by and subtract the result from the original dividend .
Step 4: Repeat the process with the new polynomial obtained after subtraction, dividing the leading term by to find the next term of the quotient.
Step 5: Continue this process until the degree of the remainder is less than the degree of the divisor, then express the remainder as a partial fraction decomposition.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Long Division of Polynomials
Long division of polynomials is a method used to divide a polynomial by another polynomial of lower degree. It involves dividing the leading term of the dividend by the leading term of the divisor, multiplying the entire divisor by this result, and subtracting it from the dividend. This process is repeated until the degree of the remainder is less than the degree of the divisor.
Partial fraction decomposition is a technique used to express a rational function as a sum of simpler fractions. This is particularly useful when integrating rational functions. The process involves factoring the denominator and expressing the function as a sum of fractions whose denominators are the factors of the original denominator, allowing for easier manipulation and integration.
The Remainder Theorem states that when a polynomial f(x) is divided by a linear divisor of the form (x - c), the remainder of this division is f(c). In the context of polynomial long division, the remainder can be expressed as a polynomial of lower degree than the divisor, which is essential for performing partial fraction decomposition and understanding the overall division process.