Textbook Question
Simplify each rational expression. Assume all variable expressions represent positive real numbers. (Hint: Use factoring and divide out any common factors as a first step.) [2(2x-3)1/3 - (x-1)(2x-3)-2/3] / [(2x-2)-2/3]
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Ch. R - Review of Basic Concepts
Lial13th EditionCollege AlgebraISBN: 9780136881063
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Table of contents
Chapter 1, Problem 100
Perform each division. See Examples 9 and 10.
Verified step by step guidance1
Identify the given expression to divide: \(\frac{3x^3 - 2x + 5}{x - 3}\). This is a polynomial division problem where the numerator is a cubic polynomial and the denominator is a linear polynomial.
Set up the polynomial long division by writing \(3x^3 - 2x + 5\) (the dividend) under the division bar and \(x - 3\) (the divisor) outside the division bar.
Divide the leading term of the dividend, \$3x^3\(, by the leading term of the divisor, \)x\(, to find the first term of the quotient. This gives \)3x^2$.
Multiply the entire divisor \(x - 3\) by this term \$3x^2$ and subtract the result from the dividend. Then bring down the next terms to form a new polynomial.
Repeat the process: divide the leading term of the new polynomial by \(x\), multiply the divisor by this result, subtract, and continue until the degree of the remainder is less than the degree of the divisor.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Polynomial Long Division
Polynomial long division is a method used to divide a polynomial by another polynomial of lower degree, similar to numerical long division. It involves dividing the leading term of the dividend by the leading term of the divisor, multiplying, subtracting, and repeating until the remainder has a lower degree than the divisor.
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Introduction to Polynomials
Degree of a Polynomial
The degree of a polynomial is the highest power of the variable in the expression. Understanding the degree helps determine the steps in division and when to stop, as the division process continues until the remainder's degree is less than the divisor's degree.
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Remainder and Quotient in Polynomial Division
When dividing polynomials, the result consists of a quotient and possibly a remainder. The quotient is the polynomial obtained from the division, and the remainder is what is left over, having a degree less than the divisor. The original expression can be written as divisor × quotient + remainder.
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Related Practice
Textbook Question
Add or subtract as indicated. 3.87 + 15 + 2.9
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Textbook Question
Let U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13}, M = {0, 2, 4, 6, 8}, N = {1, 3, 5, 7, 9, 11, 13}, Q = {0, 2, 4, 6, 8, 10, 12}, and R = {0, 1, 2, 3, 4}.Use these sets to find each of the following. Identify any disjoint sets. (R ∪ N) ∩ M′
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Textbook Question
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Textbook Question
Simplify each expression. Write answers without negative exponents. Assume all variables represent positive real numbers. (x2/3)2/(x2)7/3
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Textbook Question
Perform the indicated operations. Assume all variables represent positive real numbers. ∜32 + 3∜2
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