What is a Multiple of an Algebraic Expression?

S3-SA1-0640

Grade Level:

Class 7

AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering

Definition

What is it?

A multiple of an algebraic expression is what you get when you multiply that expression by any whole number (like 1, 2, 3, etc.) or even another algebraic expression. It's like finding multiples of a normal number, but with letters and numbers together. Essentially, it's the result of multiplying the original expression by some 'factor'.

Simple Example

Quick Example

Imagine you have an algebraic expression like '2x'. If you multiply '2x' by 3, you get '6x'. So, '6x' is a multiple of '2x'. Similarly, if you multiply '2x' by 'y', you get '2xy', which is also a multiple of '2x'.

Worked Example

Step-by-Step

Let's find some multiples of the algebraic expression '3a + 2'.

Step 1: Multiply the expression by 2.
(3a + 2) * 2 = (3a * 2) + (2 * 2) = 6a + 4

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Step 2: Multiply the expression by 5.
(3a + 2) * 5 = (3a * 5) + (2 * 5) = 15a + 10

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Step 3: Multiply the expression by 'b'.
(3a + 2) * b = (3a * b) + (2 * b) = 3ab + 2b

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Answer: Some multiples of '3a + 2' are '6a + 4', '15a + 10', and '3ab + 2b'.

Why It Matters

Understanding multiples of algebraic expressions is key for solving complex problems in science and technology. Engineers use this concept when designing structures or circuits, and computer scientists apply it in algorithms for data analysis. It helps build a strong foundation for future careers in AI/ML and Physics.

Common Mistakes

MISTAKE: Multiplying only one part of the expression. For example, for 2(x+3), writing 2x+3. | CORRECTION: Remember to multiply the number outside the bracket by EACH term inside the bracket. So, 2(x+3) = 2x + 6.

MISTAKE: Confusing multiples with factors. For example, thinking '2x' is a multiple of '4x'. | CORRECTION: A multiple is larger (or more complex) than the original expression. '4x' is a multiple of '2x' because 2x * 2 = 4x. '2x' is a FACTOR of '4x'.

MISTAKE: Ignoring variables during multiplication. For example, for x(x+y), writing x+xy. | CORRECTION: When multiplying variables, remember exponent rules. x * x = x^2. So, x(x+y) = x^2 + xy.

Practice Questions

Try It Yourself

QUESTION: Is '10y + 15' a multiple of '2y + 3'? | ANSWER: Yes (because (2y + 3) * 5 = 10y + 15)

QUESTION: Find two different multiples of the expression '4p - q'. | ANSWER: (Answers may vary) 8p - 2q (by multiplying by 2); 12p - 3q (by multiplying by 3); 4pr - qr (by multiplying by r)

QUESTION: If '6m + 9n' is a multiple of '2m + 3n', what did you multiply '2m + 3n' by? | ANSWER: 3

MCQ

Quick Quiz

Which of the following is a multiple of the expression '5x - 2y'?

5x - 2y + 3

10x - 4y

5x + 2y

x - y

The Correct Answer Is:

Option B, '10x - 4y', is obtained by multiplying the original expression '5x - 2y' by 2. Options A, C, and D are not results of multiplying the original expression by a single factor.

Real World Connection

In the Real World

Imagine a mobile data plan where the cost depends on how many GBs you use. If the base cost is 'C' rupees for 'G' GBs, and you need a plan that offers 3 times the data and 3 times the base cost, you're essentially finding a multiple of your original data expression. This helps telecom companies design various plans for customers, just like how different 'multiples' of an expression give different outcomes.

Key Vocabulary

Key Terms

MULTIPLE: The result of multiplying an expression by a number or another expression | EXPRESSION: A combination of numbers, variables, and operation signs | TERM: A single number, variable, or product of numbers and variables | FACTOR: A number or expression that divides another number or expression exactly

What's Next

What to Learn Next

Great job understanding multiples! Next, you should explore 'Factors of Algebraic Expressions'. This will help you learn how to break down complex expressions into simpler parts, which is super useful for solving equations and simplifying problems.