Domanda del libro di testo
Use the Leading Coefficient Test to determine the end behavior of the graph of the polynomial function.
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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 21
In Exercises 17–24, a) List all possible rational roots. b) List all possible rational roots. c) Use the quotient from part (b) to find the remaining roots and solve the equation. 6x3+25x2−24x+5=0
Guida verificata passo dopo passo1
Identify the polynomial given: \(6x^{3} + 25x^{2} - 24x + 5 = 0\).
For part (a), list all possible rational roots using the Rational Root Theorem. The possible roots are of the form \(\pm \frac{p}{q}\), where \(p\) divides the constant term (5) and \(q\) divides the leading coefficient (6). So, list all factors of 5 and 6, then form all possible fractions.
For part (b), test each possible rational root from part (a) by substituting into the polynomial or using synthetic division to find which one is an actual root. Once a root is found, perform synthetic division to divide the polynomial by \((x - r)\), where \(r\) is the root found, to get the quotient polynomial.
For part (c), use the quotient polynomial from part (b), which will be a quadratic, and solve it using factoring, completing the square, or the quadratic formula to find the remaining roots.
Combine all roots found (the rational root from part (b) and the roots from the quadratic in part (c)) to write the complete solution set for the equation.
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Rational Root Theorem
The Rational Root Theorem helps identify all possible rational roots of a polynomial equation by considering factors of the constant term and the leading coefficient. These possible roots are expressed as ±(factors of constant term) / (factors of leading coefficient). This theorem narrows down candidates for testing actual roots.
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Rational Exponents
Polynomial Division (Synthetic or Long Division)
Polynomial division is used to divide a polynomial by a binomial of the form (x - r), where r is a root. This process simplifies the polynomial to a lower degree, making it easier to find remaining roots. Synthetic division is a shortcut method often used for this purpose.
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07:30Introduction to Factoring Polynomials
Solving Polynomial Equations
After factoring or dividing the polynomial, solving the resulting lower-degree polynomial or quadratic equation involves finding roots using factoring, quadratic formula, or other algebraic methods. These roots, combined with previously found roots, give the complete solution set.
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5:02Solving Logarithmic Equations
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