Problem 64 - 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 $$ t^{4}=A^{2} $$

Step-by-Step Solution

Answer

Answer: - For a solution of t=0, A must be 0. - For a positive solution, A can be any non-zero value. - There is no value of A that will yield a negative solution.

Step 1: Investigate the behavior of the equation

First, we notice that the equation \(t^4 = A^2\) involves the variables \(t\) and \(A\). The variable \(t\) is raised to the power of 4 which indicates that any non-zero value of \(t\) will result in a positive outcome for \(t^4\). Thus, \(t=0\) is the only value of \(t\) that can produce \(t^4=0\).

Step 2: Determine the values of A for which t=0 is a solution

Knowing that \(t^4\) will only be equal to zero if \(t=0\), we can analyze the equation under this condition. Using the given equation: $$ t^4 = A^2 $$ and plugging in \(t=0\), we get $$ 0^4 = A^2 $$ This equation simplifies to \(0 = A^2\). From this, we can conclude that if \(A=0\), the given equation will have a solution \(t=0\).

Step 3: Determine the values of A for which the equation has a positive solution

Since \(t^4\) is always non-negative (i.e., positive or zero), any non-negative value of \(A^2\) will yield a non-negative value of \(t^4\). Hence, in order to find a positive solution for the given equation, we should look for cases where \(A^2 > 0\). Using the given equation: $$ t^4 = A^2 $$ we can deduce that if \(A \neq 0\), the equation will have a positive solution.

Step 4: Determine the values of A for which the equation has a negative solution

As mentioned earlier, \(t^4\) is always non-negative for any real value of \(t\). This implies that no value of \(A\) can make the given equation yield a negative solution for \(t^4\).

Step 5: Summarize the results

In summary, for the given equation \(t^4 = A^2\), the values of the constant \(A\) are as follows: - The equation has a solution \(t=0\) if \(A=0\). - The equation has a positive solution if \(A \neq 0\). - The equation has no negative solution for any value of \(A\).

Key Concepts

Polynomial Equations

Polynomial equations are algebraic equations involving polynomials, which are expressions composed of variables raised to non-negative integer powers. A polynomial can have one or more terms. In our exercise, the polynomial equation presented is \(t^4 = A^2\). Here, the variable \(t\) is raised to the power of 4, and the expression \(A^2\) acts as a constant term even though \(A\) can change. The degree of the polynomial is the highest power of the variable, which is 4 in this case. This makes it a quartic equation as it includes a term \(t^4\). Understanding the degree is key because it suggests the number of possible solutions. A degree 4 polynomial can have up to 4 real or complex solutions, considering multiplicities. In simpler terms, it could mean two pairs of real or imaginary roots. However, when squaring both sides as shown in \(A^2\), it indicates a specific type of symmetry and reduces the complexity of finding solutions.

Solutions of Equations

Finding the solution to an equation means determining the values of the variable that make the equation true. In the case of \(t^4 = A^2\), we are interested in finding values of \(t\) for different constants \(A\) that satisfy the equation.

• For \(t = 0\), solving gives \(A = 0\), since \(0^4 = A^2\) implies \(0 = A^2\).
• To attain a positive solution for \(t\), any non-zero value of \(A\) suffices, meaning \(A eq 0\).
• As \(t^4\) is always non-negative, negative solutions are not possible for any real values of \(A\).

Equations like these help illuminate how different values of constants can entirely change the set of solutions possible. Quadratic, cubic, and higher degree polynomials each present their own unique characteristics when solving for variable values.

Variables and Constants

Variables and constants are foundational elements in algebraic equations. In the exercise equation \(t^4 = A^2\), \(t\) represents the variable, while \(A\) functions as a constant.

• Variables are symbols that can change and stand for unknown or varying quantities. Here, \(t\) is the value we aim to find.
• Constants are fixed values that remain the same throughout the problem-solving process. In our context, \(A\) is a constant whose value we discover affects the solution set.

Understanding how these elements interact is crucial to solving equations. When \(A\) changes, it can transform the nature of the equation's solutions. Hence, constants don’t merely remain static; their values dictate the behavior of the variables involved.