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Ch. 3 - Polynomial and Rational Functions
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 3 - Polynomial and Rational FunctionsProblem 45
Chapter 4, Problem 45

Solve the equation 12x3+16x2−5x−3=0 given that -3/2 is a root.

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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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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

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. f(x)=x4−2x3+x2+12x+8

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

An equation of a quadratic function is given. a) Determine, without graphing, whether the function has a minimum value or a maximum value. b) Find the minimum or maximum value and determine where it occurs. c) Identify the function's domain and its range. f(x)=5x2−5x

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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.

Textbook Question

Describe in words the variation shown by the given equation. z=kxy2z = \(\frac{k\sqrt{x}\)}{y^2}

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