Problem 18 - 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 \(\pm 2,3\)

Step-by-Step Solution

Answer

The polynomial is \( f(x) = x^3 - 3x^2 - 4x + 12 \).

Step 1: Identify the polynomial form

The polynomial is of degree 3, which means it can be expressed in the form \( f(x) = a(x - r_1)(x - r_2)(x - r_3) \) where \( r_1, r_2, r_3 \) are the zeros.

Step 2: Substitute zeros into the polynomial

We have zeros \( r_1 = 2, \) \( r_2 = -2, \) and \( r_3 = 3 \). Substitute these into the polynomial: \( f(x) = a(x - 2)(x + 2)(x - 3) \).

Step 3: Set the leading coefficient to 1

The problem states that the leading coefficient must be 1, so set \( a = 1 \). The polynomial becomes \( f(x) = (x - 2)(x + 2)(x - 3) \).

Step 4: Expand the polynomial

First, expand \( (x - 2)(x + 2) \): \( (x - 2)(x + 2) = x^2 - 4 \). Then expand by multiplying the result by \( (x - 3) \): \((x^2 - 4)(x - 3) = x^3 - 3x^2 - 4x + 12 \). Thus, \( f(x) = x^3 - 3x^2 - 4x + 12 \).

Key Concepts

Zeros of Polynomials

Zeros of a polynomial are values of the variable that make the whole polynomial equal to zero. In simpler terms, if you plug in a zero into the polynomial, the result will be zero. For our exercise, these zeros are given as \(+2\), \(-2\), and \(3\). These zeros are essential because they can help us build the polynomial by placing them in the factors. The zeros are translated into factors in the form of \((x-r)\), where \(r\) is a zero. So, for the zeros \(+2\), \(-2\), and \(3\), the factors become \((x-2)\), \((x+2)\), and \((x-3)\). Replace the values into the polynomial, and you start forming a foundational structure of the polynomial from these zeros. Knowing the zeros is very useful because they provide us with the solutions, or roots of the polynomial, when setting the equation to equal zero. This knowledge can be harnessed in solving more complex algebraic equations.

Degree of Polynomials

The degree of a polynomial refers to the highest power of the variable within the polynomial expression. It dictates the fundamental shape and the number of zeros or roots the polynomial can hold.In this exercise, the polynomial is of degree 3, which hints that it should feature a highest power of \(x^3\). Therefore, before expanding, you can predict that the polynomial has up to three zeros or roots. You'll often hear that understanding the degree helps describe the behavior of the polynomial, especially as the variable approaches infinity. The degree also informs us of the polynomial's potential graph complexity, such as a cubic polynomial, displaying certain inflection points that are characteristic of its specific degree. Knowing the degree of a polynomial can also aid in factoring and solving it.

Expanding Polynomials

Expanding polynomials involves expressing the polynomial in an expanded form from its factored or multiplied components. It requires multiplying the factors together and simplifying the expression as necessary.In our exercise, you start with the polynomial in its factored form: \((x - 2)(x + 2)(x - 3)\). To expand this, you first focus on \((x - 2)(x + 2)\), which simplifies to \(x^2 - 4\). Once simplified, you then multiply the result \((x^2 - 4)\) by \((x - 3)\). Carefully distribute each term of the binomial across each term of the polynomial \(x^2 - 4\). When expanded fully, it turns into \(x^3 - 3x^2 - 4x + 12\), providing a straightforward polynomial with a clear leading coefficient and all expected terms visible. Expanding is crucial for understanding the polynomial's behavior and further calculus applications like differentiation or integration.