What is LCM of Polynomials by Factorisation? | Simple Explanation for Class 7

S3-SA1-0644

What is the LCM of Polynomials by Factorisation?

Grade Level:

Class 7

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

Definition

What is it?

The LCM (Least Common Multiple) of polynomials by factorisation is the smallest polynomial that is a multiple of all the given polynomials. We find it by breaking down each polynomial into its simplest factors, just like finding LCM for numbers.

Simple Example

Quick Example

Imagine two friends, Rohan and Priya, buy different types of sweets. Rohan buys sweets in packs of (x+1) pieces and Priya buys in packs of (x+2) pieces. If they want to find the smallest number of sweets that can be bought by both of them in whole packs, they would be looking for the LCM of (x+1) and (x+2).

Worked Example

Step-by-Step

Let's find the LCM of polynomials (x^2 + 5x + 6) and (x^2 + 4x + 4).

1. Factorise the first polynomial: x^2 + 5x + 6
We look for two numbers that multiply to 6 and add to 5. These are 2 and 3.
So, x^2 + 5x + 6 = (x + 2)(x + 3)

2. Factorise the second polynomial: x^2 + 4x + 4
This is a perfect square trinomial (a+b)^2 = a^2 + 2ab + b^2.
Here, a=x and b=2. So, x^2 + 4x + 4 = (x + 2)^2 = (x + 2)(x + 2)

3. List all unique factors from both polynomials: (x + 2) and (x + 3).

4. For each unique factor, take the highest power it appears in either polynomial.
(x + 2) appears as (x + 2) in the first polynomial and as (x + 2)^2 in the second. The highest power is (x + 2)^2.
(x + 3) appears as (x + 3) in the first polynomial. The highest power is (x + 3).

5. Multiply these highest powers together to get the LCM.
LCM = (x + 2)^2 (x + 3)

Answer: The LCM is (x + 2)^2 (x + 3).

Why It Matters

Understanding LCM of polynomials is crucial in fields like Computer Science and Engineering for optimising code and designing efficient systems. It helps in simplifying complex expressions in Physics calculations and is used in Cryptography to secure digital information. This skill can lead to careers in software development or data analysis.

Common Mistakes

MISTAKE: Not fully factorising the polynomials before finding the LCM. | CORRECTION: Always factorise each polynomial completely into its prime factors first, like breaking down 12 into 2x2x3.

MISTAKE: Taking only common factors once, instead of their highest powers. | CORRECTION: For each unique factor, pick the one with the highest power from any of the given polynomials.

MISTAKE: Multiplying all factors from both polynomials without considering common ones. | CORRECTION: Identify all unique factors and then take the highest power of each unique factor, then multiply them.

Practice Questions

Try It Yourself

QUESTION: Find the LCM of (x + 4) and (x^2 + 8x + 16). | ANSWER: (x + 4)^2

QUESTION: Find the LCM of (x^2 - 9) and (x + 3). | ANSWER: (x - 3)(x + 3)

QUESTION: Find the LCM of (x^2 - 4x) and (x^2 - 16). | ANSWER: x(x - 4)(x + 4)

MCQ

Quick Quiz

What is the LCM of (x + 5) and (x^2 - 25)?

(x + 5)

(x - 5)

(x + 5)(x - 5)

(x + 5)^2 (x - 5)

The Correct Answer Is:

First, factorise x^2 - 25 as (x - 5)(x + 5). The other polynomial is (x + 5). The unique factors are (x + 5) and (x - 5), each with power 1. So, the LCM is (x + 5)(x - 5).

Real World Connection

In the Real World

Imagine you are designing a game in India where characters move based on polynomial equations. To ensure all characters reach a common meeting point or complete a task at the same time, you might need to find the LCM of their movement equations. This helps in synchronizing actions and animations in games.

Key Vocabulary

Key Terms

POLYNOMIAL: An expression of more than two algebraic terms, especially the sum of several terms that contain different powers of the same variable.| FACTORISATION: The process of breaking down a polynomial into a product of simpler polynomials.| LEAST COMMON MULTIPLE (LCM): The smallest positive number that is a multiple of two or more given numbers (or polynomials).| TERM: A single number or variable, or numbers and variables multiplied together.

What's Next

What to Learn Next

Great job on understanding LCM of polynomials! Next, you can explore "HCF of Polynomials by Factorisation." Knowing both LCM and HCF will help you solve more complex problems involving algebraic fractions and equations.