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 55

Chapter 4, Problem 55

Find a polynomial function ƒ(x) of degree 3 with real coefficients that satisfies the given conditions. Zeros of -2, 1, and 0; ƒ(-1)=-1

Verified step by step guidance

Identify the zeros of the polynomial function. Since the zeros are -2, 1, and 0, the factors of the polynomial are $(x + 2)$, $(x - 1)$, and $x$ respectively.

Write the general form of the polynomial function as $f(x) = a \cdot x \cdot (x + 2) \cdot (x - 1)$, where $a$ is a real number coefficient to be determined.

Use the given condition $f(-1) = -1$ to find the value of $a$. Substitute $x = -1$ into the polynomial: $f(-1) = a \cdot (-1) \cdot (-1 + 2) \cdot (-1 - 1)$.

Simplify the expression from the substitution to get an equation in terms of $a$: $f(-1) = a \cdot (-1) \cdot 1 \cdot (-2) = 2a$.

Set the simplified expression equal to the given value $-1$ and solve for $a$: $2a = -1$, then find $a$. Once $a$ is found, write the final polynomial function $f(x)$ by substituting $a$ back into the general form.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

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 consisting of variables and coefficients combined using addition, subtraction, and multiplication, with non-negative integer exponents. The degree of a polynomial is the highest exponent of the variable, which determines the general shape and number of roots of the function. For this problem, the polynomial must be cubic (degree 3).

Zeros of a Polynomial

Zeros (or roots) of a polynomial are the values of x for which the function equals zero. If a polynomial has real coefficients and zeros at -2, 1, and 0, then (x + 2), (x - 1), and x are factors of the polynomial. The polynomial can be expressed as a product of these factors, possibly multiplied by a constant.

Using a Point to Find the Leading Coefficient

Given a point on the polynomial, such as ƒ(-1) = -1, you can substitute x = -1 into the factored form to solve for the unknown leading coefficient. This step ensures the polynomial not only has the correct zeros but also passes through the specified point, fully determining the function.

Recommended video:

Guided course

04:27

Finding Equations of Lines Given Two Points

Related Practice

Textbook Question

For each polynomial function, identify its graph from choices A–F. ƒ(x)=(x-2)²(x-5)²

Textbook Question

Solve each rational inequality. Give the solution set in interval notation. (2x + 3)/(x - 5) ≤ 0

views

Textbook Question

For each polynomial function, identify its graph from choices A–F.

$ƒ (x) = (x - 2) (x - 5)$

Textbook Question

Identify any vertical, horizontal, or oblique asymptotes in the graph of y=ƒ(x). State the domain of ƒ.

views

Textbook Question

Use synthetic division to determine whether the given number k is a zero of the polynomial function. If it is not, give the value of ƒ(k). ƒ(x) = 5x⁴ + 2x³ -x+3; k=2/5

Textbook Question

Solve each rational inequality. Give the solution set in interval notation. (3x + 7)/(x - 3) ≤ 0

views