Problem 66 - Elementary and Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

The value of the polynomial is 31.

Step 1: Substitute the Given Values

Substitute the given values of \(a = -2\) and \(b = 3\) into the polynomial \(a^3 - 2ab + b^3\). The expression becomes:\[(-2)^3 - 2(-2)(3) + (3)^3\]

Step 2: Calculate the First Term

Calculate \((-2)^3\). This is \(-8\), since \[(-2) \times (-2) \times (-2) = -8\]So, the expression now is:\[-8 - 2(-2)(3) + (3)^3\]

Step 3: Calculate the Second Term

Calculate \(-2(-2)(3)\). This is equal to \(12\), since \[-2 \times (-2) = 4\] and \[4 \times 3 = 12\]So, the expression now is:\[-8 + 12 + (3)^3\]

Step 4: Calculate the Third Term

Calculate \((3)^3\). This is equal to \(27\), since \[3 \times 3 \times 3 = 27\]The expression now is:\[-8 + 12 + 27\]

Step 5: Simplify the Expression

Add the terms together:\(-8 + 12 + 27 = 31\)

Key Concepts

Substitution Method

Polynomial evaluation often begins with the **substitution method**, a straightforward approach where you replace variables with specific values. Think of it as swapping letters for numbers. In our original exercise, the polynomial is given as \(a^3 - 2ab + b^3\), and we're asked to evaluate it for \(a = -2\) and \(b = 3\). By substituting these values, the polynomial transforms.

The original expression becomes:

• \((-2)^3\) for \(a^3\)
• \(-2(-2)(3)\) for \(-2ab\)
• \((3)^3\) for \(b^3\)

The key is to carefully replace each variable while maintaining the mathematical structure. This process allows us to transform abstract expressions into numerical ones, making them easier to solve.

Simplifying Expressions

**Simplifying expressions** involves breaking down complex expressions into more manageable parts by performing operations such as addition, subtraction, multiplication, and division. After substituting the specific values of \(a\) and \(b\) into the polynomial, you simplify by resolving these operations step by step. For example, after substitution, we're left with:

• \[-8 - 2(-2)(3) + (3)^3\]

Simplification happens in stages:
1. Resolve each individual term, such as \((-2)^3\), \(-2(-2)(3)\), and \((3)^3\).
2. Combine the simplified values by following the order of operations: first handle exponents, then multiplications and divisions, and finally additions and subtractions.

In our exercise, careful simplification yields \(-8 + 12 + 27\), further reduced to 31 by direct addition. Simplifying expressions is crucial as it ensures clarity and accuracy in your calculations.

Exponents Calculation

Understanding **exponents calculation** is essential when evaluating expressions involving powers. Exponents tell you how many times to multiply a number by itself. In our specific polynomial, expressions like \((-2)^3\) and \((3)^3\) feature prominently.

To calculate \((-2)^3\):

• Multiply \((-2) \times (-2)\) to get 4.
• Then, 4 multiplied by \(-2\) results in \(-8\).

Similarly, for \((3)^3\):

• Multiply \(3 \times 3\) for 9.
• Then, multiply 9 by 3 to get 27.

Handling exponents accurately is critical as they can significantly alter the outcome of calculations. Remember, each step adheres to the rules of signs: multiplying a negative number results in a negative, and positive numbers remain positive. Mastering this ensures you approach polynomial evaluation with confidence and precision.