Problem 16 - Algebra Form and Function

Question

In Exercises \(1-21,\) solve the equation for the variable. $$ \frac{1}{4} t^{3}=\frac{4}{t} $$

Step-by-Step Solution

Answer

Question: Solve the equation \(\frac{1}{4}t^3 = \frac{4}{t}\) for variable \(t\). Answer: The solutions for the given equation are \(t = 2\) and \(t = -2\).

Step 1: Eliminate the fractions

To eliminate the fractions, we need to find the LCM, which in this case is \(4t\). Multiply both sides of the equation by \(4t\) to eliminate the fractions: $$ (4t) \left( \frac{1}{4} t^{3} \right) = (4t) \left( \frac{4}{t} \right) $$

Step 2: Simplify the equation

Simplify both sides of the equation after multiplying with LCM: $$ t^{4} = 16 $$

Step 3: Isolate the variable t

We have a simple polynomial equation \(t^4 = 16\). To solve for t, we need to take the fourth root on both sides of the equation: $$ t = \pm \sqrt[4]{16} $$

Step 4: Calculate the solution(s) for t

Find the fourth root of 16 and consider both positive and negative values as solutions: $$ t = \pm 2 $$ So the solutions for the given equation are \(t = 2\) and \(t = -2\).

Key Concepts

Fraction Elimination in Equations

Eliminating fractions in equations is a fundamental skill in algebra. The presence of fractions can make solving equations more cumbersome. Thus, the first step is to eliminate them. You do this by multiplying each term of the equation by the least common multiple (LCM) of the denominators that appear in the equation.
For instance, in the equation \( \frac{1}{4} t^{3} = \frac{4}{t} \), the LCM of the denominators \( 4 \) and \( t \) is \( 4t \). By multiplying each side by \( 4t \), you remove the fractions from the equation:

• Multiply \( (4t) \left( \frac{1}{4} t^{3} \right) \) to get \( t^4 \)
• Multiply \( (4t) \left( \frac{4}{t} \right) \) to get \( 16 \)

This results in the simplified equation \( t^4 = 16 \). Eliminating fractions simplifies solving equations, making it a vital skill to master.

Understanding Polynomial Equations

Polynomial equations are equations involving a polynomial expression in terms of a variable. A polynomial is a sum of terms with variables raised to whole number powers and coefficients. In this context, \( t^4 = 16 \) is a polynomial equation where the highest power of \( t \) is 4.
Polynomial equations are named by their degree, which is the highest exponent of the variable. Here, it's called a "quartic" equation because of its degree of 4.
Solving polynomial equations often requires isolating the variable. This might involve factoring, using the quadratic formula, or performing operations like taking roots. In this example, you isolate \( t \) by computing the fourth root on both sides:

• \( t^4 = 16 \)
• \( t = \pm \sqrt[4]{16} \)

Understanding polynomials and how to solve them will equip you with techniques to tackle more complex algebraic problems.

Finding the Roots of Equations

The roots of an equation are the solutions that make the equation true. They can be interpreted as the x-values where the graph of a polynomial crosses the x-axis. To find the roots, you solve for the variable.
In the equation \( t^4 = 16 \), you look for values of \( t \) that satisfy the equation. This process often involves extracting roots. Here, it requires taking the fourth root of both sides. Remember that both positive and negative values are valid for even roots:

• The fourth root of 16 is 2.
• Thus, \( t = 2 \) and \( t = -2 \) are both roots of the given equation.

These are real numbers indicating the points where the polynomial equation equals zero. Finding roots is essential as it helps solve equations by pinpointing their solutions.