Problem 17 - Precalculus: Functions and Graphs

Question

Find a polynomial \(f(x)\) with leading coefficient 1 and having the given degree and zeros. degree \(3 ; \quad\) zeros \(-2,0,5\)

Step-by-Step Solution

Answer

The polynomial is \(f(x) = x^3 - 3x^2 - 10x\).

Step 1: Identify the structure of the polynomial

Since the polynomial has a degree of 3 and the leading coefficient should be 1, the polynomial can be expressed in the form \( f(x) = (x - r_1)(x - r_2)(x - r_3) \), where \( r_1, \ r_2, \ r_3 \) are the zeros of the polynomial.

Step 2: Substitute the zeros into the polynomial

The zeros given are \(-2, 0,\) and \(5\). Substitute these zeros into the expression from Step 1: \( f(x) = (x + 2)(x)(x - 5) \).

Step 3: Expand the polynomial

First, expand \((x + 2)(x)\) to get \(x^2 + 2x\), and then multiply this result by \((x - 5)\) to get the complete polynomial:\[(x^2 + 2x)(x - 5) = x^3 - 5x^2 + 2x^2 - 10x = x^3 - 3x^2 - 10x\].

Key Concepts

Degree of a Polynomial

When we talk about the degree of a polynomial, we're looking at the highest power of the variable, usually denoted as 'x,' in the polynomial expression. In simple terms, the degree tells us how many times the variable is multiplied by itself in the term with the highest exponent. For instance, in the polynomial \(x^3 - 3x^2 - 10x\), the degree is 3.

• The degree of a polynomial is a non-negative integer.
• It tells us the maximum number of solutions or roots a polynomial can have.
• It also indicates the number of ways the graph of the polynomial can cross the x-axis, which we call the zeros.

Understanding the degree is essential when forming polynomials. In our exercise, we started by knowing the polynomial's degree and computed it to be 3, meaning the highest power of 'x' in our polynomial equation is \(x^3\). This prepares the ground for finding the polynomial and its specific zeros.

Zeros of a Polynomial

Zeros of a polynomial are the values of 'x' that make the polynomial equal to zero. In simple terms, they are the 'solutions' or 'roots' of the polynomial equation. For example, if you have a polynomial equation \(f(x) = (x + 2)(x)(x - 5)\), the zeros are the solutions where \(f(x) = 0\).

• A polynomial of degree 3 can have up to 3 zeros.
• Each zero represents a point where the graph of the polynomial intersects with the x-axis.
• Zeros are critical since they can guide us in factoring a polynomial.

In our task, we started with given zeros \(-2, 0, \) and \(5\). By understanding that a polynomial's zeros can be written as factors \((x - r_1)(x - r_2)(x - r_3)\), we can easily substitute the zeros into these factors to form the factored version of the polynomial. This step directs us towards simplifying and understanding the polynomial further.

Leading Coefficient

The leading coefficient is the coefficient of the term with the highest degree in a polynomial. It's a key feature of any polynomial since it affects the shape and direction of the graph. For instance, in the polynomial \(x^3 - 3x^2 - 10x\), the leading coefficient is 1, which is in front of \(x^3\) - the term with the highest degree.

• If the leading coefficient is positive, the right end of the graph of the polynomial tends to rise.
• If it's negative, the right end of the graph tends to fall.
• The leading coefficient also affects the rate of growth or reduction of the polynomial values as 'x' increases or decreases.

For our polynomial, which had a specified leading coefficient of 1, this means the polynomial would have an even symmetry about the origin, assuming no integer restrictions. It ensures our expansion follows the rules given for constructing the polynomial and keeps the graph's characteristics stable as determined by this coefficient.