Textbook Question
Find the vertical asymptotes, if any, and the values of x corresponding to holes, if any, of the graph of each rational function. r(x)=x/(x2+4)
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Ch. 3 - Polynomial and Rational Functions
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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Table of contents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Chapter 4, Problem 27
Divide using long division.
Verified step by step guidance1
Set up the long division by writing the dividend \$4x^3 - 3x^2 - 2x + 1\( under the division bar and the divisor \)x + 1$ outside the division bar.
Divide the leading term of the dividend, \$4x^3\(, by the leading term of the divisor, \)x\(, to get the first term of the quotient: \)4x^2$.
Multiply the entire divisor \(x + 1\) by \$4x^2\( to get \)4x^3 + 4x^2$, then subtract this from the dividend to find the new remainder.
Bring down the next term from the original dividend and repeat the process: divide the leading term of the new remainder by \(x\), multiply the divisor by this result, subtract, and continue.
Continue this process until all terms have been brought down and divided, resulting in the quotient and possibly a remainder expressed over 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 the divisor by this result, subtracting from the dividend, and repeating the process until the remainder has a lower degree than the divisor.
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Introduction to Polynomials
Leading Terms and Degree of Polynomials
The leading term of a polynomial is the term with the highest exponent, and the degree is the exponent of that term. Understanding these helps determine the first step in division and when to stop the process, as division continues until the remainder's degree is less than the divisor's degree.
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Remainder and Quotient in Polynomial Division
In polynomial division, the quotient is the result of the division, and the remainder is what is left when the division cannot continue. The remainder must have a degree less than the divisor. The division can be expressed as Dividend = Divisor × Quotient + Remainder.
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Related Practice
Textbook Question
Find an nth-degree polynomial function with real coefficients satisfying the given conditions. If you are using a graphing utility, use it to graph the function and verify the real zeros and the given function value. n=3; -5 and 4+3i are zeros; f(2) = 91
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Textbook Question
Divide using synthetic division. (x7+x5−10x3+12)/(x+2)
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Textbook Question
Divide using synthetic division. (x5+x3−2)/(x−1)
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Textbook Question
Use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola's axis of symmetry. Use the graph to determine the function's domain and range. f(x)=x2−2x−3
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Textbook Question
Solve each polynomial inequality in Exercises 1–42 and graph the solution set on a real number line. Express each solution set in interval notation. 9x2−6x+1<0
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