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Ch. 3 - Polynomial and Rational Functions
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 3 - Polynomial and Rational FunctionsProblem 24
Chapter 4, Problem 24

In Exercises 19–24, use the Leading Coefficient Test to determine the end behavior of the graph of the polynomial function. f(x)=−11x4−6x2+x+3f(x)=−11x^4−6x^2+x+3

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Identify the leading term of the polynomial function. The leading term is the term with the highest power of x, which in this case is −11x4.
Determine the degree of the polynomial, which is the exponent of the leading term. Here, the degree is 4, an even number.
Look at the leading coefficient, which is the coefficient of the leading term. In this function, the leading coefficient is -11, a negative number.
Apply the Leading Coefficient Test: For an even degree polynomial, if the leading coefficient is negative, the end behavior of the graph is that both ends go down as x approaches positive and negative infinity.
Summarize the end behavior: As x→∞, f(x)→−∞, and as x→−∞, f(x)→−∞.

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Key Concepts

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

Leading Coefficient Test

The Leading Coefficient Test uses the degree and leading coefficient of a polynomial to determine its end behavior. For large positive or negative x-values, the sign and degree dictate whether the graph rises or falls on each end. This test simplifies predicting the graph's behavior without plotting points.
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End Behavior of Polynomial Functions

Degree of a Polynomial

The degree of a polynomial is the highest power of the variable in the expression. It indicates the general shape and number of turning points of the graph. Even-degree polynomials have similar end behaviors on both sides, while odd-degree polynomials have opposite end behaviors.
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Standard Form of Polynomials

Leading Coefficient

The leading coefficient is the coefficient of the term with the highest degree in a polynomial. Its sign (positive or negative) affects the direction of the graph's ends. A positive leading coefficient causes the graph to rise on the right end, while a negative one causes it to fall.
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Related Practice
Textbook Question

Divide using synthetic division. (x5+4x4−3x2+2x+3)÷(x−3)

Textbook Question

Solve each polynomial inequality in Exercises 1–42 and graph the solution set on a real number line. Express each solution set in interval notation. −x2 + 2x ≥ 0

Textbook Question

Find an nth-degree polynomial function with real coefficients satisfying the given conditions. If you are using a graphing utility, use it to graph the function and verify the real zeros and the given function value. n=3; 1 and 5i are zeros; f(-1) = -104

Textbook Question

Divide using synthetic division. (6x5−2x3+4x2−3x+1)÷(x−2)

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Textbook Question

Find the zeros for each polynomial function and give the multiplicity for each zero. State whether the graph crosses the x-axis, or touches the x-axis and turns around, at each zero. f(x)=2(x−5)(x+4)2

Textbook Question

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. x4−2x3−5x2+8x+4=0

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