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What is the Factorisation of Polynomials?
Grade Level:
Class 6
AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering
Definition
What is it?
Factorisation of Polynomials is like breaking down a big number into smaller numbers that multiply together to give the original number. Here, we break down an algebraic expression (called a polynomial) into simpler expressions (called factors) that multiply together to form the original polynomial.
Simple Example
Quick Example
Imagine you have a box of 10 laddoos. You can arrange them as 2 rows of 5 laddoos (2 x 5) or 5 rows of 2 laddoos (5 x 2). Here, 2 and 5 are factors of 10. Similarly, for a polynomial like x^2 + 5x, we can write it as x(x + 5). Here, x and (x + 5) are the factors.
Worked Example
Step-by-Step
Let's factorise the polynomial: 3x + 6
1. Look for a common number or variable that divides both terms (3x and 6).
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2. The number 3 divides both 3x (giving x) and 6 (giving 2).
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3. So, 3 is the common factor.
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4. Write the common factor outside a bracket: 3( )
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5. Inside the bracket, write what's left after dividing each term by the common factor: (3x / 3) + (6 / 3)
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6. This simplifies to x + 2.
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7. So, the factorised form is 3(x + 2).
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Answer: 3(x + 2)
Why It Matters
Factorisation helps simplify complex problems, just like simplifying a big fraction. Engineers use it to design bridges and buildings, and computer scientists use it to write efficient code for apps and games. It's a fundamental tool in solving many real-world challenges.
Common Mistakes
MISTAKE: Not finding the 'greatest' common factor, e.g., factorising 4x + 8 as 2(2x + 4) | CORRECTION: Always find the largest possible common factor. Here, 4 is the greatest common factor, so it should be 4(x + 2).
MISTAKE: Forgetting a term inside the bracket after dividing, e.g., factorising 5x + 5 as 5(x) | CORRECTION: Remember to divide ALL terms by the common factor. 5x + 5 should be 5(x + 1) because 5 divided by 5 is 1.
MISTAKE: Incorrectly handling signs, e.g., factorising -2x - 4 as -2(x - 2) | CORRECTION: When taking out a negative common factor, all signs inside the bracket flip. -2x - 4 should be -2(x + 2).
Practice Questions
Try It Yourself
QUESTION: Factorise 7y + 14 | ANSWER: 7(y + 2)
QUESTION: Factorise 5x^2 + 10x | ANSWER: 5x(x + 2)
QUESTION: Factorise 6ab - 9ac | ANSWER: 3a(2b - 3c)
MCQ
Quick Quiz
Which of the following is the correct factorisation of 4p - 8?
2(2p - 4)
4(p - 2)
4p - 8
8(p - 1)
The Correct Answer Is:
B
Option B is correct because 4 is the greatest common factor of both 4p and 8. When you divide 4p by 4 you get p, and when you divide -8 by 4 you get -2. Options A, C, and D are incorrect or incomplete factorisations.
Real World Connection
In the Real World
When you use a music streaming app like Spotify or JioSaavn, algorithms suggest songs you might like. These algorithms use mathematical operations, including polynomial factorisation, to find patterns in your listening habits and other users' preferences. It helps them group similar songs and recommend them efficiently.
Key Vocabulary
Key Terms
POLYNOMIAL: An expression with one or more terms, like x^2 + 2x + 1 | FACTOR: A number or expression that divides another number or expression exactly | TERM: Parts of an expression separated by + or - signs, e.g., in 3x + 6, '3x' and '6' are terms | COMMON FACTOR: A factor that is shared by two or more terms.
What's Next
What to Learn Next
Great job learning about factorisation! Next, you can explore 'Algebraic Identities'. These are special equations that are always true and will make factorisation even faster and easier for more complex polynomials.
