Ch. 3 - Polynomial and Rational Functions

Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook

Lial 13th Edition

Ch. 3 - Polynomial and Rational Functions

Problem 37

Chapter 4, Problem 37

Solve each problem. Find a polynomial function ƒ of degree 3 with -2, 1, and 4 as zeros, and ƒ(2)=16.

Verified step by step guidance

Start by writing the general form of a cubic polynomial function with the given zeros. Since the zeros are -2, 1, and 4, the function can be expressed as: \[f(x) = a(x + 2)(x - 1)(x - 4)\] where $a$ is a constant coefficient to be determined.

Expand the factors $(x + 2)(x - 1)(x - 4)$ step-by-step. First, multiply two of the binomials, for example $(x + 2)(x - 1)$, and then multiply the result by the third binomial $(x - 4)$.

After expanding, you will have a cubic polynomial in the form $a(x^3 + bx^2 + cx + d)$. Keep the coefficient $a$ as a variable since it is not yet known.

Use the given condition $f(2) = 16$ to find the value of $a$. Substitute $x = 2$ into the expanded polynomial and set the expression equal to 16. This will give you an equation in terms of $a$.

Solve the equation for $a$, then write the final polynomial function $f(x)$ by substituting the value of $a$ back into the expanded polynomial.

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Key Concepts

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

Polynomial Functions and Degree

A polynomial function is an expression involving variables raised to whole-number exponents and coefficients. The degree of a polynomial is the highest exponent of the variable, which determines the function's general shape and number of roots. For a degree 3 polynomial, there are up to three zeros or roots.

Zeros (Roots) of a Polynomial

Zeros of a polynomial are the values of the variable that make the function equal to zero. If -2, 1, and 4 are zeros, then (x + 2), (x - 1), and (x - 4) are factors of the polynomial. The polynomial can be expressed as a product of these factors multiplied by a constant.

Using a Point to Find the Leading Coefficient

Given a point on the polynomial, such as ƒ(2) = 16, you can substitute x = 2 into the factored form to solve for the leading coefficient. This constant scales the polynomial so it passes through the given point, ensuring the function satisfies all conditions.

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Textbook Question

For each polynomial function, one zero is given. Find all other zeros. $ƒ (x) = - x^{4} - 5 x^{2} - 4; - i$

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Textbook Question

Current Flow In electric current flow, it is found that the resistance offered by a fixed length of wire of a given material varies inversely as the square of the diameter of the wire. If a wire 0.01 in. in diameter has a resistance of 0.4 ohm, what is the resistance of a wire of the same length and material with diameter 0.03 in., to the nearest ten-thousandth of an ohm?

Textbook Question

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

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) = (x + 3)²

Textbook Question

Graph each polynomial function. Factor first if the polynomial is not in factored form. See Examples 3 and 4.

$ƒ (x) = x^{3} + x^{2} - 36 x - 36$

Textbook Question

Give the equations of any vertical, horizontal, or oblique asymptotes for the graph of each rational function. ƒ(x)=3/(x-5)

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