Problem 64 - Elementary and Intermediate Algebra

Question

Evaluate each polynomial for \(a=-2\) and \(b=3 .\) See Example 4. $$ a^{3}-b^{3} $$

Step-by-Step Solution

Answer

The value of the polynomial is -35.

Step 1: Identify the Polynomial

The polynomial we need to evaluate is \(a^3 - b^3\). We are given the values \(a = -2\) and \(b = 3\). Our goal is to substitute these values into the polynomial and simplify.

Step 2: Substitute Values

Substitute \(a = -2\) and \(b = 3\) into the polynomial. This gives us \((-2)^3 - (3)^3\).

Step 3: Calculate Powers

Calculate \((-2)^3\) and \(3^3\): \((-2)^3 = -2 \times -2 \times -2 = -8\) and \(3^3 = 3 \times 3 \times 3 = 27\).

Step 4: Subtract the Results

After substituting and calculating the powers, we have the expression \(-8 - 27\). Subtracting these values gives us \(-8 - 27 = -35\).

Key Concepts

Substitution Method

The substitution method is a technique used in mathematics to replace variables with their given numerical values. Here, you deal with a polynomial and directly substitute the given values into the designated spots within the equation. Let’s look at a simple example with the polynomial \( a^3 - b^3 \). Given \( a = -2 \) and \( b = 3 \), you’ll substitute these values into the expression, replacing every instance of \( a \) and \( b \) with \(-2\) and \(3\), respectively.

• The first step is to locate all instances of the variable.
• Then, carefully replace each variable with its corresponding number.
• This step sets the stage for further evaluation like calculating exponents or carrying out arithmetic operations.

The substitution method is mainly about maintaining accuracy in replacing numbers to ensure correct evaluation. Simple errors in substitution can lead to incorrect conclusions, so practice being meticulous through each step.

Exponents

Exponents are a mathematical tool used to represent repeated multiplication of a number by itself. In our polynomial, both \(a^3\) and \(b^3\) involve exponents, where the number is multiplied by itself three times. Evaluating \((-2)^3\) and \(3^3\) involves the following steps:

• For \((-2)^3\), multiply \(-2\) by itself three times: \(-2 \times -2 \times -2\). This calculation gives \(-8\), as two negatives give a positive, and the third negative turns it back negative.
• For \(3^3\), you multiply \(3\) by itself three times: \(3 \times 3 \times 3\). This results in \(27\). Here, each number is positive, so the result remains positive.

Understanding how to calculate exponents is crucial to solving polynomials accurately, especially since incorrect exponentiation will lead to wrong answers. Be sure to practice different cases, including negative bases and large exponents, to enhance your understanding.

Simplification

Simplification in the context of evaluating polynomials involves reducing an expression to its simplest form after performing all calculations like substitution and exponentiation. After substitution and calculating \((-2)^3\) and \(3^3\), the expression \(-8 - 27\) is evaluated next by subtraction.

When simplifying:

• First, ensure all calculations have been done correctly up to this point.
• In our example, subtraction of the two numbers \(-8\) and \(27\) (notice both these numbers are similarly negative in result after prior calculations) is straightforward: \(-8 - 27\) equals \(-35\).
• The final answer should be clear of unneeded complexity.

The goal of simplification is to reach the most basic form of an answer, making it easy to interpret and understand. Simplification also serves to confirm the accuracy of previous steps, solidifying one's understanding of the solution.