Problem 7 - College Algebra with Corequisite Support

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 \( -2, 3, \) and \( -5 \).

Step 1: Identify the Equation

We start with the polynomial function given by \( C(t) = 3(t+2)(t-3)(t+5) \). Here, \( t \) represents the variable, similar to \( x \) in other contexts.

Step 2: Understanding Intercepts

The \( t \)-intercepts occur where the polynomial is equal to zero, \( C(t) = 0 \). This means finding values of \( t \) that make the equation \( C(t) = 3(t+2)(t-3)(t+5) \) equal to zero.

Step 3: Set the Equation to Zero

To find the intercepts, we set the polynomial equal to zero: \[ 3(t+2)(t-3)(t+5) = 0 \] Since 3 is a nonzero constant, we can ignore it to focus on the factors involving \( t \).

Step 4: Solve for Each Factor

Set each factor equal to zero and solve for \( t \): 1. \( t+2=0 \) leads to \( t=-2 \)2. \( t-3=0 \) leads to \( t=3 \)3. \( t+5=0 \) leads to \( t=-5 \)

Step 5: List the Intercepts

The \( t \)-intercepts of the polynomial are the values of \( t \) calculated in the previous step: \( -2, 3, \) and \( -5 \). These are the points where the function crosses the \( t \)-axis.

Key Concepts

X-Intercepts

Understanding x-intercepts is key to analyzing polynomial functions. In simple terms, x-intercepts are the points where the graph of a polynomial crosses the x-axis. Since the y-coordinate at any x-intercept is zero, the x-intercepts are the roots of the equation when the polynomial is set to zero.

To find x-intercepts, follow these steps:

• Set the polynomial equation equal to zero.
• Solve for the variable, typically x. This involves finding the values of x that satisfy the equation.

Remember, for a polynomial function, such as one structured like our given polynomial function in terms of t, the x-intercepts are analogous to the t-intercepts. Therefore, learning how to detect these intercepts enhances our understanding of how the polynomial behaves graphically.

T-Intercepts

The concept of t-intercepts is closely related to x-intercepts, but applies when the variable in the polynomial is t instead of x. The t-intercepts occur where the graph of the polynomial crosses the t-axis, meaning the value of the function is zero at these points.

For the function provided, the intercepts can be found by examining where the polynomial equals zero: \( C(t) = 3(t+2)(t-3)(t+5) = 0 \). When this is set to zero, each factor of the polynomial must be zero individually.

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

Thus, the t-intercepts for the polynomial are at t = -2, 3, and -5. These points provide insights into where the polynomial graph interacts with the t-axis.

Factoring Polynomials

Factoring polynomials is a fundamental skill in finding intercepts and simplifying complex equations. It involves breaking down a polynomial into simpler terms (factors), which, when multiplied together, give the original polynomial.

The provided function \( C(t) = 3(t+2)(t-3)(t+5) \) is a perfect illustration of factoring. Here is how factoring helps:

• When the polynomial is completely factored, as in \( 3(t+2)(t-3)(t+5) \), each binomial factor like \( (t+2) \), \( (t-3) \), and \( (t+5) \) represents potential solutions or zeros.
• If each factor is set to zero, solving these simpler equations quickly yields the roots or intercepts of the polynomial.

Factoring is not only essential for finding intercepts but also plays a critical role in calculus and higher-level mathematics. It reveals the structure of a polynomial and highlights its zeros, aiding in graph sketching and providing insights into function behavior.