Problem 7 - College Algebra

Question

For the following exercises, find the \(x\) - or \(t\) -intercepts of the polynomial functions. $$ C(t)=3(t+2)(t-3)(t+5) $$

Step-by-Step Solution

Answer

The \(t\)-intercepts are \((-2,0)\), \((3,0)\), and \((-5,0)\).

Step 1: Understand the Problem

We need to find the \(t\)-intercepts of the polynomial function \(C(t)=3(t+2)(t-3)(t+5)\). \(t\)-intercepts occur where the function crosses the \(t\)-axis, which means where the function value is zero, or \(C(t) = 0\).

Step 2: Set the Polynomial to Zero

To find the \(t\)-intercepts, set the polynomial equal to zero: \[ 3(t+2)(t-3)(t+5) = 0 \].

Step 3: Apply the Zero-Product Property

The Zero-Product Property states that if a product of factors is zero, at least one of the factors must be zero. Set each factor equal to zero:1. \(t+2=0\)2. \(t-3=0\)3. \(t+5=0\).

Step 4: Solve Each Equation

Solve each of the three equations for \(t\):For \(t+2=0\), solve to get \(t=-2\).For \(t-3=0\), solve to get \(t=3\).For \(t+5=0\), solve to get \(t=-5\).

Step 5: Conclude the Solution

The solutions \(t = -2\), \(t = 3\), and \(t = -5\) are the \(t\)-intercepts of the polynomial function \(C(t)\). These intercepts correspond to the points \((-2, 0)\), \((3, 0)\), and \((-5, 0)\) on the \(t\)-axis.

Key Concepts

Zero-Product Property

The Zero-Product Property is a fundamental principle in algebra that simplifies finding the roots of polynomial equations. In essence, this property states that if the product of two or more factors equals zero, then at least one of the factors must be zero. This concept is critical when solving polynomial functions like the one in our exercise: \[ C(t) = 3(t + 2)(t - 3)(t + 5) = 0 \]To find the intercepts, we set the polynomial equal to zero and apply the Zero-Product Property. This involves setting each factor of \((t + 2)(t - 3)(t + 5)\) equal to zero. It allows us to break down a complex equation into simpler, manageable parts:

• \(t + 2 = 0\)
• \(t - 3 = 0\)
• \(t + 5 = 0\)

By solving each equation separately, we find the specific values of \(t\) that satisfy the equation, which are the intercepts of the polynomial function. This property is particularly useful because it eliminates the need for trial and error, letting us systematically find solutions.

Polynomial Functions

Polynomial functions are expressions involving variables and coefficients, connected with operations of addition, subtraction, multiplication, and non-negative integer exponents. They are a central topic in algebra, providing a versatile tool to model various phenomena. The general form of a polynomial function can be expressed as:\[ P(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 \]In our example, the polynomial function is \(C(t) = 3(t + 2)(t - 3)(t + 5)\). Notice how it's distinctly factorized, which is particularly helpful in identifying intercepts easily. The degree of a polynomial function refers to the highest power of the variable, which in this case is 3 (since it would expand to a cubic term), implying there can be up to three real roots or intercepts. Understanding polynomial functions is essential for grasping concepts like derivative functions, rate of change, and curve sketching.Polynomial equations are not only easy to solve when in factorized form but also provide insights into the shape and behavior of their graphs on the coordinate plane. Each root is where the graph will cross the axis, giving us the intercepts.

Intercepts in Algebra

Intercepts are key points in algebra where a graph crosses the axes. Specifically, the \(x\)-intercepts (or \(t\)-intercepts if the variable is \(t\)) are where the function graph intersects the horizontal axis. Finding these intercepts is about identifying the values that make the equation zero.In practical terms, for the given polynomial function \(C(t)=3(t+2)(t-3)(t+5)\):

• The intercepts are found where \(C(t) = 0\).
• This happens at \(t = -2\), \(t = 3\), and \(t = -5\).

Each intercept describes a specific point on the graph, such as \((-2, 0)\), \((3, 0)\), and \((-5, 0)\). Intercepts play a vital role in understanding the geometry of the graph, assisting in sketching, and analyzing behavior of functions over different intervals.Knowing the intercepts provides insight into the turning points and symmetries of the graph. By consistently applying strategies to find both \(x\)-and \(y\)-intercepts, students can better predict and understand how functions behave graphically, leading to deeper mathematical insights.