Problem 67 - Algebra Form and Function

Question

In Problems \(59-70,\) decide for what values of the constant \(A\) the equation has (a) The solution \(t=0\) (b) A positive solution (c) A negative solution $$ t^{3}+1=A $$

Step-by-Step Solution

Answer

Question: Determine the values of the constant A such that the equation t^3 + 1 = A has (a) a solution of t=0, (b) a positive solution, and (c) a negative solution. Answer: (a) For a solution of t=0, A=1. (b) For a positive solution, A > 1. (c) For a negative solution, A < 1.

Step 1: Case (a): t=0

First, we'll consider the case where the equation has a solution where t=0. We'll plug in t=0 and solve for A. $$ (0)^3 + 1 = A $$ This simplifies to: $$ A = 1 $$ So, for A=1, the equation has a solution of t=0.

Step 2: Case (b): Positive Solution

Now, let's consider the case where the equation has a positive solution, i.e., t > 0. For t > 0, we know that t^3 > 0. Thus, when adding 1 to both sides, we have: $$ t^3 + 1 > 0 + 1 $$ Which simplifies to: $$ A > 1 $$ So, for A > 1, the equations have a positive solution.

Step 3: Case (c): Negative Solution

Finally, let's consider the case where the equation has a negative solution, i.e., t < 0. For t < 0, we know that t^3 < 0. Then, adding 1 to both sides, we get: $$ t^3 + 1 < 0 + 1 $$ Which simplifies to: $$ A < 1 $$ So, for A < 1, the equations have a negative solution.

Key Concepts

Cubic Equations

Cubic equations are mathematical expressions of the form \( t^3 + b t^2 + c t + d = 0 \). They involve a variable that is raised to the third power, or cubed. In the context of our exercise, the specific cubic equation is \( t^3 + 1 = A \). Cubic equations can have one, two, or three real solutions, depending on the specific values of the coefficients and the constant term. Solving cubic equations often requires understanding and applying various methods such as factoring, the rational root theorem, or utilizing synthetic division. However, when simplified to the form \( t^3 + 1 = A \), we focus on how the constant \( A \) affects the solutions of the equation. Knowing these properties helps to predict and verify the number and type of solutions possible in cubic equations.

Solution Sets

A solution set is a collection of values that satisfy an equation. For the cubic equation \( t^3 + 1 = A \), determining the solution set involves finding which values of \( t \) meet the condition set by a particular \( A \).

• If \( A = 1 \), the solution set includes \( t = 0 \).
• If \( A > 1 \), possible solutions include positive values of \( t \), since the equation requires \( t^3 + 1 \) to be a larger number than one.
• If \( A < 1 \), the equation accommodates negative values of \( t \), because \( t^3 + 1 \) needs to be less than one.

Understanding the solution set helps in verifying that the equation behaves as expected under various conditions.

Constants

Constants in algebra are fixed values that do not change. They are crucial in determining the behavior of equations and their solutions. In our case, the constant \( A \) greatly influences the cubic equation \( t^3 + 1 = A \).

• For \( A = 1 \), the equation finds a special solution at \( t=0 \), making the equation perfectly balanced.
• Contrastingly, raising the value of \( A \) above 1 leads to the potential for positive \( t \) solutions.
• Lowering \( A \) below 1 allows for negative \( t \) solutions to satisfy the equation.

Analyzing how a constant like \( A \) affects an equation gives insights into how subtle changes can lead to different mathematical results.

Inequalities

Inequalities define the relationship between expressions that are not equal but show one is greater or less than the other. In our exercise, inequalities help explore the potential solutions of the equation \( t^3 + 1 = A \) when \( A \) does not precisely equal 1.

• For \( A > 1 \), the inequality \( t^3 + 1 > 0 + 1 \) indicates that only positive \( t \) would satisfy the equation.
• Conversely, when \( A < 1 \), the inequality \( t^3 + 1 < 0 + 1 \) points to negative \( t \) values as solutions.

Understanding inequalities enriches comprehension of how equations change under different conditions and which sets of solutions are feasible. They are fundamental in describing ranges of possible values that satisfy both sides of an equation and help in predicting behavior of real-world scenarios translated into mathematical terms.