Problem 28 - Elementary Algebra

Question

Find the value of each expression for the following problems. $$ P=n(n-1)(n-2) . \text { Find } P \text { if } n=-3 $$

Step-by-Step Solution

Answer

Answer: When \(n = -3\), the value of \(P\) is 60.

Step 1: Substitute the value of n into the expression

Given the expression \(P = n(n-1)(n-2)\) and the value of n being -3, we will substitute -3 into the expression: $$ P = (-3)((-3) - 1)((-3) - 2) $$

Step 2: Simplify the expression

Now, we will simplify the expression by performing the operations inside the parentheses first. $$ P = (-3)(-4)(-5) $$

Step 3: Calculate the value of P

Finally, we will multiply the numbers together to find the value of \(P\). $$ P = 3 \cdot 4 \cdot 5 = 60 $$ So, the value of \(P\) when \(n = -3\) is 60.

Key Concepts

Substitution in Algebra

In algebra, substitution is a straightforward method used to replace a variable with a specific value. This technique is particularly useful when you need to calculate the outcome of an algebraic expression for a given variable. For example, consider the expression \( P = n(n-1)(n-2) \). If you are told that \( n = -3 \), then substitution involves replacing every occurrence of \( n \) in the expression with \( -3 \).
To apply substitution correctly:

• Identify the variable to be replaced in the expression.
• Replace each instance of the variable with the provided value, ensuring accurate insertion inside any parentheses or alongside any operators.

This will set the stage for further simplification and computation, as seen when substituting \( n = -3 \) in our example, leading to \( P = (-3)((-3) - 1)((-3) - 2) \). Ensuring accurate substitution is essential for obtaining the correct result.

Simplifying Expressions

Simplification in algebra involves reducing an expression to its simplest form. The goal is to make the expression as concise and straightforward as possible, while preserving its original value. This often involves combining like terms, performing arithmetic operations, or simplifying fractions.
For our expression \( (-3)((-3) - 1)((-3) - 2) \), simplification involves resolving the operations inside the parentheses first. Calculate each bracket:

• \((-3) - 1 = -4\)
• \((-3) - 2 = -5\)

Thus, the expression becomes \( (-3)(-4)(-5) \). It's crucial to follow the order of operations (often remembered as PEMDAS/BODMAS) when simplifying any algebraic expression to ensure accuracy.

The Process of Multiplication

Multiplication is one of the core operations in mathematics, essential for combining numbers through repeated addition. When dealing with the expression \((-3)(-4)(-5)\), multiplication is used to combine these values into a single product.
Break down the multiplication as follows:

• Multiply the first two numbers: \((-3) \times (-4) = 12\)
• Then multiply the result by the next number: \(12 \times (-5) = -60\)

It is important to remember the rules of multiplication, especially with negative numbers:

• The product of two negative numbers is positive.
• The product of a positive and a negative number is negative.

Thus, while \((-3) \times (-4)\) yields a positive outcome, further multiplying by \((-5)\) results in a negative product, \(-60\). Understanding these rules helps prevent mistakes in calculations.