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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 35
Chapter 4, Problem 35

Graph each polynomial function. Factor first if the polynomial is not in factored form. ƒ(x)=(3x-1)(x+2)2

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Identify the given polynomial function: \(f(x) = (3x - 1)(x + 2)^2\). Since it is already factored, we can proceed to analyze it for graphing.
Determine the zeros of the function by setting each factor equal to zero: solve \(3x - 1 = 0\) and \(x + 2 = 0\) to find the x-intercepts.
Analyze the multiplicity of each zero: the factor \((3x - 1)\) has multiplicity 1, and \((x + 2)^2\) has multiplicity 2. This affects the behavior of the graph at these points.
Find the y-intercept by evaluating \(f(0)\): substitute \(x = 0\) into the function and simplify to get the y-coordinate where the graph crosses the y-axis.
Determine the end behavior of the polynomial by considering the degree and leading coefficient: since \((x + 2)^2\) is squared and \(3x - 1\) is linear, multiply their leading terms to find the overall degree and leading coefficient, which will guide how the graph behaves as \(x\) approaches positive and negative infinity.

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Key Concepts

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

Polynomial Functions

Polynomial functions are expressions consisting of variables raised to whole-number exponents and their coefficients. Understanding their general shape and behavior helps in graphing, especially recognizing how degree and leading coefficients affect the graph's end behavior.
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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 reveals the roots or zeros of the function, which are critical points where the graph intersects the x-axis.
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Introduction to Factoring Polynomials

Graphing Using Zeros and Multiplicities

The zeros of a polynomial correspond to x-intercepts on the graph. The multiplicity of each zero determines the graph's behavior at that point: odd multiplicities cross the x-axis, while even multiplicities touch and turn around.
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Finding Zeros & Their Multiplicity
Related Practice
Textbook Question

Solve each problem. Is x+1 a factor of ƒ(x)=x3+2x2+3x+2?

Textbook Question

For each polynomial function, one zero is given. Find all other zeros. ƒ(x)=4x3+6x2−2x−1; 12ƒ(x)=4x^3+6x^2-2x-1;\(\text{ }\]\frac\)12

Textbook Question

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Textbook Question

Graph each polynomial function. Factor first if the polynomial is not in factored form. ƒ(x)=(4x+3)(x+2)2

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Textbook Question

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

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