Problem 60 - Basic Technical Mathematics with Calculus

Question

Solve the given problems. In finding the volume \(V\) (in \(\mathrm{cm}^{3}\) ) of a certain gas in equilibrium with a liquid, it is necessary to solve the equation \(V^{3}-6 V^{2}+12 V=8\) Use synthetic division to determine if \(V=2 \mathrm{cm}^{3}\)

Step-by-Step Solution

Answer

\(V = 2\, \mathrm{cm}^3\) is a solution, as the remainder is zero.

Step 1: Set up Synthetic Division

To check if \(V=2\) is a solution to the equation \(V^3 - 6V^2 + 12V = 8\), rewrite the equation as \(V^3 - 6V^2 + 12V - 8 = 0\). Use synthetic division with the root \(V = 2\). Set up the synthetic division by writing 2 outside and the coefficients 1, -6, 12, and -8 inside (representing \(V^3 - 6V^2 + 12V - 8\)).

Step 2: Perform Synthetic Division

Bring down the first coefficient, which is 1. Multiply it by 2 and place the result, 2, under the second coefficient (-6). Add -6 and 2 to get -4. Multiply -4 by 2 to get -8, and place it under the next coefficient (12). Add 12 and -8 to get 4. Multiply 4 by 2 to get 8, and place it under the last coefficient (-8). Add -8 and 8 to get 0.

Step 3: Interpret the Remainder

The remainder from the synthetic division is 0. This indicates that \(V = 2\) is indeed a root of the polynomial equation \(V^3 - 6V^2 + 12V - 8 = 0\). Since the remainder is 0, \(2\) is a solution to the equation.

Key Concepts

Polynomial Equation

A polynomial equation is an expression that involves a polynomial set equal to a value, often zero. In this exercise, the polynomial equation is formed by the expression:

• \( V^3 - 6V^2 + 12V = 8 \).

To solve problems involving polynomial equations, it's common to first rearrange the equation such that one side is zero. This allows us to better identify methods like synthetic division for finding specific solutions. In this case, rewriting it gives:

• \( V^3 - 6V^2 + 12V - 8 = 0 \).

Polynomial equations can range from simple linear equations to more complex ones involving higher degrees, like this cubic (degree 3) polynomial. Solving such equations can often reveal multiple solutions or roots, guiding us to understand more about the function's behavior.

Roots of Polynomials

The roots of a polynomial are the values of the variable that make the polynomial equal to zero. They represent the points where the polynomial intercepts the x-axis in a graphical representation. For example, if we plug a root back into the polynomial, the output becomes zero.
In the context of our exercise, we check if \( V = 2 \) is a root by substituting it into the equation and simplifying. To find this out systematically, we can use synthetic division to divide the polynomial by \( V - 2 \):

• Bringing down the first coefficient and multiplying through the steps gives a remainder.
• If the remainder is zero, then \( V=2 \) is a root.

In this exercise, the remainder was indeed zero, confirming \( V = 2 \) is a valid root. The number of roots a polynomial has is equal to its degree in a fundamental theorem of algebra context, and understanding the roots helps reveal characteristics of the polynomial function.

Polynomial Remainder Theorem

The Polynomial Remainder Theorem is a handy mathematical tool. It states that for any polynomial \( f(x) \) divided by \( x - c \), the remainder is \( f(c) \). This is powerful because it allows us to check for roots efficiently.
For instance, if \( V = 2 \) is to be tested as a root of our polynomial \( V^3 - 6V^2 + 12V - 8 \), by using synthetic division, we determine what \( f(2) \) would be. Setting up synthetic division and working through the coefficients, we check:

• The remainder after division was 0.
• This confirms \( V=2 \) as a root, as per the theorem's prediction.

Thus, the Polynomial Remainder Theorem simplifies the process of finding or verifying roots without needing to fully evaluate the polynomial's factors directly. It’s a time-effective way to confirm solutions to polynomial equations, especially when dealing with higher-degree polynomials.