Textbook Question
Use the four-step procedure for solving variation problems given on page 447 to solve Exercises 1–10. y varies directly as x and inversely as the square of z. y = 20 when x = 50 and z = 5. Find y when x = 3 and z = 6.
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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 5
Divide using long division. State the quotient, and the remainder, .
Verified step by step guidance1
Set up the long division by writing the dividend \(6x^{3} + 7x^{2} + 12x - 5\) under the division bar and the divisor \(3x - 1\) outside the division bar.
Divide the leading term of the dividend \$6x^{3}\( by the leading term of the divisor \)3x$ to find the first term of the quotient: \(\frac{6x^{3}}{3x} = 2x^{2}\).
Multiply the entire divisor \(3x - 1\) by the term \$2x^{2}$ and subtract the result from the dividend to find the new polynomial to bring down.
Repeat the process: divide the new leading term by \$3x$, multiply the divisor by this result, subtract, and continue until the degree of the remainder is less than the degree of the divisor.
Express the final answer as the quotient polynomial plus the remainder over the divisor, in the form \(\text{quotient} + \frac{r(x)}{3x - 1}\).
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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
Quotient and Remainder in Polynomial Division
When dividing polynomials, the quotient is the polynomial result of the division, and the remainder is the leftover polynomial with a degree less than the divisor. The division can be expressed as Dividend = Divisor × Quotient + Remainder.
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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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Related Practice
Textbook Question
Use the Rational Zero Theorem to list all possible rational zeros for each given function. f(x)=x5−x4−7x3+7x2−12x−12
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Textbook Question
Use the Rational Zero Theorem to list all possible rational zeros for each given function. f(x)=4x4−x3+5x2−2x−6
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Textbook Question
Divide using long division. State the quotient, and the remainder, .
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Textbook Question
In Exercises 1–8, use the Rational Zero Theorem to list all possible rational zeros for each given function. f(x)=3x4−11x3−3x2−6x+8
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Textbook Question
Find the domain of each rational function. h(x)=(x+7)/(x2−49)
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