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Factorisation Problems – FAQs with Detailed Solutions
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Factorisation is an important topic in algebra that students learn in Classes 7 to 10. It is the method of breaking down a number or a polynomial into smaller parts called factors. These factors, when multiplied together, give back the original number or expression. For example, the factorisation of the expression x² + 2x is x(x + 2). Here, x and (x + 2) are the factors.
Learning how to factorise helps in solving equations easily and is very useful in higher mathematics. In this topic, you will learn how to find common terms, use identities, and rearrange terms to break down complex expressions. In this article, we’ll go through some solved factorisation problems to help you understand the steps clearly. You’ll also find a few practice questions at the end so you can test how well you’ve understood the concept.
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Solving Factorisation Problems
Q.1: Factorise 6x² + 19x + 10
Solution:
We break the middle term 19x into two parts:
→ 6x² + 15x + 4x + 10
Now, take out common terms from each pair:
→ 3x(2x + 5) + 2(2x + 5)
Take out the common factor:
→ (3x + 2)(2x + 5)
Q.2: Factorise z² + 13z + 30
Solution:
Split the middle term 13z into 10z and 3z:
→ z² + 10z + 3z + 30
Group and factor:
→ z(z + 10) + 3(z + 10)
Final answer:
→ (z + 3)(z + 10)
Q.3: Factorise 3x² + 10x – 8
Solution:
Split the middle term 10x into 12x and -2x:
→ 3x² + 12x – 2x – 8
Take out the common terms:
→ 3x(x + 4) – 2(x + 4)
Now factor:
→ (3x – 2)(x + 4)
Q.4: Factorise 9(m + n)² – 18(m² – n²) + 16(m – n)²
Solution:
We know that m² – n² = (m + n)(m – n)
Now substitute that in:
→ 9(m + n)² – 18(m + n)(m – n) + 16(m – n)²
Now group as a perfect square:
→ [3(m + n) – 4(m – n)]²
→ [3m + 3n – 4m + 4n]²
→ (7n – m)²
Q.5: Factorise 36(p + q)² – (p – q)²
Solution:
Write the expression using squares:
→ [6(p + q)]² – (p – q)²
Use the identity a² – b² = (a + b)(a – b):
→ [6(p + q) + (p – q)] [6(p + q) – (p – q)]
Simplify both:
→ [6p + 6q + p – q] [6p + 6q – p + q]
→ [7p + 5q][5p + 7q]
Q.6: Factor 8m²n – 12mn + 6mn²
Solution:
Find the HCF, which is 2mn:
→ 2mn(4m – 6 + 3n)
Q.7: Simplify (9x² – 16y²)/(3x + 4y)
Solution:
Recognise it as a difference of squares:
→ (3x)² – (4y)² = (3x + 4y)(3x – 4y)
Now cancel (3x + 4y):
→ [(3x + 4y)(3x – 4y)] / (3x + 4y)
Final answer:
→ 3x – 4y
Additional Factorisation Practice Questions
- Factor: 20x³ + 5x² – 10x
- Factorise: 64x² − 40x + 6
- Find the factors of: 9x²yz − 7y
- Factor: x² − 13x + 36
- Factorise: 13x² + 39x − 130
- Factorise: −25 + 64x²
- Find the factors of the expression: 98 − 2x²
- Find the factors of: 16x² + 25y² + 40xy
- Factorise using algebraic identities: x² + 10x + 25
- Given that one of the factors of (6x² + 60x − 144) is (x − 3), find the other factor.
FAQs For Factorisation Problems
What is factorisation?
Factorisation is a method of factoring a number or a polynomial. The polynomials are decomposed into products of their factors.
What are the steps to solve factorisation problems?
To solve factorisation problems, first write the expression, then decompose the expression into its factors. Take out the common terms and simplify the equation to get the factors.
What is the importance of factorisation in algebra?
Factorisation is important in algebra as it helps to simplify expressions and solve complex problems. It is used to find the roots of a polynomial equation, simplify algebraic fractions, and solve word problems.
How can I practice factorisation problems?
You can practice factorisation problems by solving different types of problems given in this article. Also, you can find more practice questions at the end of the article.
Why is factorisation important in mathematics?
Factorisation helps simplify complex expressions and solve equations easily. It's essential for solving quadratic equations, simplifying algebraic fractions, and understanding polynomial identities
What are the common methods of factorisation?
Some commonly used methods include: Taking out the common factor Grouping terms Using algebraic identities Splitting the middle term Difference of squares method
How do I know which method to use when factorising?
It depends on the type and form of the expression: If all terms share a common factor → take it out. If the expression has 4 terms → try grouping. If it's a quadratic → use splitting the middle term or the quadratic formula. If it's a special form → apply an algebraic identity.
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