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Ch. 3 - Polynomial and Rational Functions
Lial - College Algebra 13th Edition
Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook
All textbooksLial 13th EditionCh. 3 - Polynomial and Rational FunctionsProblem 43
Chapter 4, Problem 43

Graph each polynomial function. Factor first if the polynomial is not in factored form. ƒ(x)=2x3-5x2-x+6

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Identify the given polynomial function: \(f(x) = 2x^3 - 5x^2 - x + 6\).
Attempt to factor the polynomial by grouping. Group the terms as \((2x^3 - 5x^2)\) and \((-x + 6)\).
Factor out the greatest common factor (GCF) from each group: from the first group factor out \(x^2\), and from the second group factor out \(-1\), giving \(x^2(2x - 5) - 1(x - 6)\).
Check if the binomials inside the parentheses are the same. If not, try rearranging or use another factoring method such as the Rational Root Theorem to find possible roots and factor accordingly.
Once factored completely, use the factored form to find the roots (zeros) of the function by setting each factor equal to zero, then plot these roots on the x-axis and analyze the end behavior to sketch the graph.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Polynomial Functions

A polynomial function is an expression consisting of variables raised to whole-number exponents and coefficients combined using addition, subtraction, and multiplication. Understanding the degree and leading coefficient helps predict the general shape and end behavior of the graph.
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Introduction to Polynomial Functions

Factoring Polynomials

Factoring involves rewriting a polynomial as a product of simpler polynomials or factors. This process helps identify the roots or zeros of the function, which correspond to the x-intercepts on the graph, making it easier to sketch the function accurately.
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Introduction to Factoring Polynomials

Graphing Polynomial Functions

Graphing a polynomial requires plotting its zeros, determining the end behavior based on degree and leading coefficient, and analyzing the function’s behavior between roots. Factoring first simplifies finding zeros, which are critical points for sketching the curve.
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Graphing Polynomial Functions
Related Practice
Textbook Question

Give the equations of any vertical, horizontal, or oblique asymptotes for the graph of each rational function. ƒ(x)=(x2-2x-3)/(2x2-x-10)

Textbook Question

Solve each polynomial inequality. Give the solution set in interval notation. x4 + 2x3 + 36 < 11x2 + 12x

Textbook Question

Graph each polynomial function. Factor first if the polynomial is not in factored form. ƒ(x)=2x4+x3-6x2-7x-2

Textbook Question

Determine the largest open interval of the domain (a) over which the function is increasing and (b) over which it is decreasing. ƒ(x) = -3x2 + 18x + 1

Textbook Question

For each polynomial function, use the remainder theorem to find ƒ(k). ƒ(x) = 2x5 - 10x3 - 19x2 - 50; k=3

Textbook Question

Determine the largest open interval of the domain (a) over which the function is increasing and (b) over which it is decreasing. ƒ(x) = -2x2 - 8x - 7

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