Problem 86 - Elementary Algebra

Question

Find the value of each of the following expressions. \(P=n(n-1)(n-2)(n-3) . \quad\) Find \(P\) if \(n=-5\)

Step-by-Step Solution

Answer

Answer: The value of the expression when \(n = -5\) is \(P = 1680\).

Step 1: Plug in the value of n

We need to find the value of \(P\) if \(n = -5\), so let's plug in this value into the expression. \(P = (-5)(-5-1)(-5-2)(-5-3)\)

Step 2: Simplify the expression

Now, let's simplify the expression by performing the operations: \(P = (-5)(-6)(-7)(-8)\)

Step 3: Multiply the numbers

All that's left is to multiply the numbers together to find the value of \(P\): \(P = 5 \times 6 \times 7 \times 8\)

Step 4: Calculate the result

Now, calculate the result: \(P = 1680\) So, the value of \(P\) when \(n = -5\) is \(1680\).

Key Concepts

Algebraic Expressions

Algebraic expressions are the backbone of algebra. They consist of variables, numbers, and operations such as addition, subtraction, multiplication, and division. In an algebraic expression, like the one in our exercise, the variable represents an unknown quantity that can be replaced by specific values.

Let's take the given example of the polynomial expression P=n(n-1)(n-2)(n-3). This expression contains the variable n, and it showcases the multiplication of four terms that are linear functions of n. In algebra, understanding how to manipulate these expressions is key to solving a wide range of problems.

Substitution Method

The substitution method is a fundamental technique in algebra. It involves replacing a variable in an algebraic expression with a given number to evaluate the expression. In our example, n is substituted with -5.

It's very important to carefully substitute the variable with the given number and remember to replace it everywhere the variable appears. After the substitution, we obtain the numerical expression (-5)(-6)(-7)(-8), which we can then evaluate through multiplication.

Negative Numbers

Negative numbers, such as -5 in our exercise, are numbers that are less than zero. They can sometimes be tricky when performing operations, especially multiplication and division.

It's crucial to remember the basic rule that multiplying two negative numbers results in a positive number. For instance, (-5) × (-6) = 30. The student should be comfortable with negative numbers to correctly evaluate expressions where they occur.

Multiplication of Integers

Multiplication of integers, both positive and negative, is governed by a set of rules that are easy to remember. Firstly, the product of two positive numbers is positive. Secondly, the product of two negative numbers is also positive, as we've seen with the negative numbers component.

When you multiply integers, you also multiply their absolute values and then determine the sign of the result following these rules. In our problem, once all integers are multiplied together as positive numbers due to the pair-wise negative multiplication, we arrive at the final positive product of 1680. Mastering these rules is essential for correct polynomial evaluation involving negative values.