Textbook Question
Give the domain and the range of each quadratic function whose graph is described. The vertex is and the parabola opens up.
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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 45
Solve the equation 12x3+16x2−5x−3=0 given that -3/2 is a root.
Verified step by step guidance1
Since \(-\frac{3}{2}\) is a root of the polynomial \(12x^{3} + 16x^{2} - 5x - 3 = 0\), use polynomial division or synthetic division to divide the cubic polynomial by the factor \(\left(x + \frac{3}{2}\right)\).
To perform synthetic division, rewrite the root \(-\frac{3}{2}\) as \(-1.5\) and set up the coefficients of the polynomial: 12, 16, -5, and -3.
Carry out the synthetic division process step-by-step to find the quotient polynomial, which will be a quadratic expression.
Once you have the quadratic quotient, set it equal to zero and solve for \(x\) using the quadratic formula: \[x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\], where \(a\), \(b\), and \(c\) are the coefficients of the quadratic.
Combine the root \(-\frac{3}{2}\) with the solutions from the quadratic to write the complete solution set for the original cubic equation.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Polynomial Roots and Factor Theorem
The Factor Theorem states that if a polynomial f(x) has a root r, then (x - r) is a factor of f(x). Given that -3/2 is a root, (x + 3/2) must be a factor of the polynomial, which helps in factoring and simplifying the equation.
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Introduction to Factoring Polynomials
Polynomial Division
Polynomial division, either long division or synthetic division, is used to divide the original polynomial by the factor corresponding to the known root. This process reduces the polynomial's degree, making it easier to solve the remaining equation.
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Solving Quadratic Equations
After factoring out the known root, the remaining polynomial is quadratic. Solving this quadratic equation using methods like factoring, completing the square, or the quadratic formula yields the other roots of the original cubic equation.
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Related Practice
Textbook Question
Solve each rational inequality in Exercises 43–60 and graph the solution set on a real number line. Express each solution set in interval notation. (x+3)/(x+4)<0
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Textbook Question
In Exercises 39–52, find all zeros of the polynomial function or solve the given polynomial equation. Use the Rational Zero Theorem, Descartes's Rule of Signs, and possibly the graph of the polynomial function shown by a graphing utility as an aid in obtaining the first zero or the first root. x4−3x3−20x2−24x−8=0
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Textbook Question
Find the horizontal asymptote, if there is one, of the graph of each rational function. f(x)=(−2x+1)/(3x+5)
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Textbook Question
Solve the equation 2x3−3x2−11x+6=0 given that -2 is a zero of f(x)=2x3−3x2−11x+6.
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Textbook Question
Describe in words the variation shown by the given equation.
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