Problem 63 - College Algebra

Question

For the following exercises, use the given information about the polynomial graph to write the equation. Degree \(3 .\) Zeros at \(x=-3, x=-2\) and \(x=1\) \(y\) -intercept at \((0,12)\)

Step-by-Step Solution

Answer

The polynomial equation is \(f(x) = -2(x + 3)(x + 2)(x - 1)\).

Step 1: Write the General Form of the Polynomial

Given that the polynomial has a degree of 3 and its zeros are at \(x = -3\), \(x = -2\), and \(x = 1\), we can write the general form of the polynomial as: \( f(x) = a(x + 3)(x + 2)(x - 1) \). Here, \(a\) is a constant that we need to determine.

Step 2: Use the y-intercept to Find 'a'

We know the \(y\)-intercept is at \((0, 12)\), which means \(f(0) = 12\). Substituting \(x = 0\) into the polynomial equation, we get \(a(0+3)(0+2)(0-1) = 12\). This simplifies to \(-6a = 12\).

Step 3: Solve for 'a'

Solve the equation \(-6a = 12\) to find \(a\). Divide both sides by \(-6\): \(a = 12 / -6 = -2\).

Step 4: Write the Final Polynomial Equation

With \(a = -2\), substitute back into the polynomial equation from Step 1 to get: \(f(x) = -2(x + 3)(x + 2)(x - 1)\). This is the equation of the polynomial.

Key Concepts

Polynomial Equations

Polynomial equations are expressions composed of variables and coefficients, connected by operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. They are at the heart of algebra and play a big role in analyzing and solving various real-life scenarios. A polynomial equation might look like this: \[ f(x) = a_nx^n + a_{n-1}x^{n-1} + \dots + a_1x + a_0 \]. Here, the terms \(a_nx^n\), \(a_{n-1}x^{n-1}\) and so forth represent coefficients \(a_n, a_{n-1}, \ldots\) multiplied by powers \(x^n, x^{n-1}, \ldots\) of the variable \(x\).

• The coefficients are real numbers.
• The exponents are whole numbers.
• The highest power of \(x\) in the polynomial determines its degree.

Understanding polynomial equations allows you to model and predict outcomes efficiently by identifying connections among numerical datasets. A key task often involves finding specific values of \(x\) satisfying the equation, also known as the "roots" or "solutions." In solving this exercise, we are required to express the relationship defined by the polynomial's given degree and zeros.

Zeros of a Polynomial

The zeros of a polynomial, also known as roots or solutions, are the values of \(x\) that make the polynomial equation equal to zero. In other words, these are the points at which the graph of the polynomial crosses the x-axis. For the polynomial \(f(x)\), it can be expressed as: \[ f(x) = a(x - r_1)(x - r_2)(x - r_3)\ldots \],where \(r_1, r_2, r_3\) are the zeros of the polynomial. Think of zeros as key landmarks on the graph that help us understand its shape and behavior.

• In this exercise, the zeros are given as \(x = -3\), \(x = -2\), and \(x = 1\).
• These zeros suggest that the polynomial can be expressed as \((x + 3)(x + 2)(x - 1)\).
• They provide valuable hints about the polynomial's factorization form.

These points are integral to solving polynomial equations and transforming data between graphical and algebraic forms, leading us to insights such as determining the 'a' constant in the problem.

Degree of a Polynomial

The degree of a polynomial is a crucial aspect that characterizes the polynomial's complexity and behavior. It is defined as the highest power of the variable \(x\) that appears in the polynomial with a non-zero coefficient. The degree provides a snapshot of potential intersections with the x-axis and gives us a sense of its growth direction and symmetry.

• For instance, in our exercise, the polynomial is described as having a degree of 3.
• This means the polynomial can have up to 3 real zeros and can have intricate behavior changes such as two direction reversals on its graph.
• The degree also determines the maximal number of turns or bends on the graph.

Understanding the degree helps us position and anticipate key features of the graph and the polynomial equation. It guides the complexity and creativity involved in exploring solutions to polynomial equations and orchestrates its overarching behavior, paving the way for comprehensive algebraic discovery.