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. 6x2+x>1
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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 14
Divide using long division. State the quotient, and the remainder, r(x). (x4+2x3−4x2−5x−6)/(x2+x−2)
Verified step by step guidance1
Identify the dividend and divisor: The dividend is \(x^{4} + 2x^{3} - 4x^{2} - 5x - 6\) and the divisor is \(x^{2} + x - 2\).
Set up the long division by writing the dividend under the division bar and the divisor outside.
Divide the leading term of the dividend \(x^{4}\) by the leading term of the divisor \(x^{2}\) to find the first term of the quotient: \(\frac{x^{4}}{x^{2}} = x^{2}\).
Multiply the entire divisor \(x^{2} + x - 2\) by \(x^{2}\) and subtract the result from the dividend to find the new polynomial to divide.
Repeat the process: divide the leading term of the new polynomial by \(x^{2}\), multiply the divisor by this term, 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 one polynomial by another, 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, and repeating until the degree of the remainder is less 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 degrees is essential in division because the process continues until the remainder's degree is less than the divisor's degree, indicating the division is complete.
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Quotient and Remainder in Polynomial Division
When dividing polynomials, the quotient is the result of the division, and the remainder is what is left over. The remainder must have a degree less than the divisor. Expressing the division as dividend = divisor × quotient + remainder helps verify the result.
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Related Practice
Textbook Question
In Exercises 9–16, a) List all possible rational zeros. b) Use synthetic division to test the possible rational zeros and find an actual zero. c) Use the quotient from part (b) to find the remaining zeros of the polynomial function. f(x)=2x3+x2−3x+1
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Textbook Question
Find the coordinates of the vertex for the parabola defined by the given quadratic function. f(x)=−x2−2x+8
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Textbook Question
Use the graph of the rational function in the figure shown to complete each statement in Exercises 15–20.
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Textbook Question
Write an equation that expresses each relationship. Then solve the equation for y. x varies directly as the cube root of z and inversely as y.
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Textbook Question
Write an equation that expresses each relationship. Then solve the equation for y. x varies jointly as y and z and inversely as the square root of w.
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