Domanda del libro di testo
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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TOC.tableOfContents
- 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
Capitolo 4, Problema 45
Solve the equation 12x3+16x2−5x−3=0 given that -3/2 is a root.
Guida verificata passo dopo passo1
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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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.
Video consigliato:
07:30
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.
Video consigliato:
05:13Introduction to Polynomials
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.
Video consigliato:
06:08Solving Quadratic Equations by Factoring
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