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Chapter 5: Problem 9

For the following exercises, find the \(x\) - or t-intercepts of the polynomial functions. $$ C(t)=2 t(t-3)(t+1)^{2} $$

Short Answer

Expert verified
The t-intercepts are \( t = 0 \), \( t = -1 \), and \( t = 3 \).

Step by step solution

01

Understand the Problem

We need to find the t-intercepts of the function \( C(t) = 2t(t-3)(t+1)^2 \). The t-intercepts occur where \( C(t) = 0 \).
02

Set the Function to Zero

To find the t-intercepts, set the polynomial function equal to zero: \( 2t(t-3)(t+1)^2 = 0 \).
03

Solve for Each Factor

Since the equation is a product of factors equal to zero, we apply the zero-product property. Set each factor equal to zero: \( 2t = 0 \), \( t-3 = 0 \), and \( (t+1)^2 = 0 \).
04

Solve Each Equation

Solve each equation separately: \( 2t = 0 \) gives \( t = 0 \); \( t-3 = 0 \) gives \( t = 3 \); \( (t+1)^2 = 0 \) gives \( t = -1 \).
05

Identify the t-intercepts

The t-intercepts are the solutions to the equations. Therefore, the t-intercepts are \( t = 0 \), \( t = -1 \), and \( t = 3 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

t-intercepts
The concept of t-intercepts in polynomial functions relates to the points where the graph of the function crosses the t-axis. These are the values of \( t \) for which the polynomial is equal to zero. In our original exercise, the goal is to identify the t-intercepts of a polynomial function given by \( C(t) = 2t(t-3)(t+1)^2 \). To find these intercepts, you need to solve for \( t \) whenever \( C(t) = 0 \).
  • t-intercepts are the solutions to the equation \( C(t) = 0 \).
  • They represent the horizontal points where the graph meets the t-axis.
  • For the function \( C(t) = 2t(t-3)(t+1)^2 \), setting \( C(t) = 0 \) initiates the process of finding these intercepts.
Understanding t-intercepts is crucial because these points provide key insights into the function's behavior and its symmetry along the t-axis.
Zero-Product Property
The Zero-Product Property is a fundamental concept in algebra, which is critical for solving polynomial equations that have been factored into products of smaller expressions. It states that if a product of two or more factors is zero, then at least one of the factors must also be zero.
  • This property allows us to solve equations like \( 2t(t-3)(t+1)^2 = 0 \) by splitting it into smaller equations: \( 2t = 0 \), \( t-3 = 0 \), \( (t+1)^2 = 0 \).
  • By applying this property, you can solve for each factor individually to find possible solutions for \( t \).
Understanding this property helps in breaking down complex polynomial equations into manageable parts, thus simplifying the process of finding solutions. This method not only aids in identifying t-intercepts but also serves as a foundational tool for higher-level algebraic tasks.
Solve Polynomial Equations
Solving polynomial equations involves finding the values of \( t \) that satisfy the equation, typically by setting the polynomial equal to zero. In the case of our exercise, this involves using both the Zero-Product Property and factoring.
  • First, represent the equation in factored form, like \( C(t) = 2t(t-3)(t+1)^2 \).
  • Set \( C(t) = 0 \) and apply the Zero-Product Property to determine each factor.
  • The factors are solved individually: \( 2t = 0 \) leads to \( t = 0 \), \( t-3 = 0 \) gives \( t = 3 \), and \( (t+1)^2 = 0 \) results in \( t = -1 \).
  • These solutions are your t-intercepts.
By breaking down the equation using these techniques, solving polynomial equations becomes more straightforward. This process highlights the importance of understanding properties like the Zero-Product Property and how they apply to finding solutions for polynomial equations.

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