Domanda del libro di testo
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. (4−2x)/(3x+4)≤0
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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 50
In Exercises 49–50, find all the zeros of each polynomial function and write the polynomial as a product of linear factors. g(x) = x^4 - 6x^3 + x^2 + 24x + 16
Guida verificata passo dopo passo1
Step 1: Use the Rational Root Theorem to list all possible rational roots of the polynomial. The possible rational roots are the factors of the constant term (16) divided by the factors of the leading coefficient (1).
Step 2: Test each possible rational root by substituting them into the polynomial \( g(x) = x^4 - 6x^3 + x^2 + 24x + 16 \) to see which ones are actual roots.
Step 3: Once a root is found, use synthetic division to divide the polynomial by \( x - \text{root} \) to reduce the polynomial's degree.
Step 4: Repeat the process of finding roots and using synthetic division on the resulting polynomial until it is reduced to a quadratic or linear factor.
Step 5: Factor the remaining quadratic polynomial, if necessary, to find all zeros and express the original polynomial as a product of linear factors.
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Polynomial Functions
A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. The general form is f(x) = a_n*x^n + a_(n-1)*x^(n-1) + ... + a_1*x + a_0, where a_n is not zero. Understanding polynomial functions is crucial for analyzing their behavior, including finding zeros, which are the values of x that make the function equal to zero.
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Introduction to Polynomial Functions
Finding Zeros
Finding the zeros of a polynomial involves determining the values of x for which the polynomial equals zero. This can be achieved through various methods, including factoring, using the Rational Root Theorem, or applying synthetic division. Zeros are essential as they indicate the x-intercepts of the graph of the polynomial and are key to expressing the polynomial as a product of linear factors.
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03:42Finding Zeros & Their Multiplicity
Factoring Polynomials
Factoring polynomials is the process of breaking down a polynomial into simpler components, specifically linear factors. For a polynomial of degree n, it can be expressed as a product of n linear factors if it has n zeros. This is important for simplifying expressions and solving equations, as well as for understanding the polynomial's roots and their multiplicities.
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07:30Introduction to Factoring Polynomials
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