Hard Level Worksheet

Part A: Subjective Questions - Very Short Answer (1 Mark Each)

This hard level explores advanced factorisation techniques including cubic identities and complex multi-term expressions.

Master these concepts to excel in competitive examinations and higher mathematics.

1. What is the difference between HCF and LCM in factorisation?

HCF is the greatest common factorhighest common factor we take out, while LCM finds least common multiplecommon multiple.

Perfect! In factorisation, we primarily use HCF to factor expressions.

2. Define difference of two squares.

Difference of two squares is a² - b² = (a + b)(a - b)an expression like a² - b².

Excellent! It's one of the most useful factorisation identities.

3. Give one example for factorisation by grouping.

Answer:

4. Factorise: x² - 16.

= (x + )(x - )

Perfect! x² - 16 = x² - 4² = (x + 4)(x - 4).

5. Factorise x² - y².

= (x + )(x - )

Correct! Using a² - b² = (a + b)(a - b).

Drag each expression to its correct factorisation category:

Part A: Section B – Short Answer Questions (2 Marks Each)

1. Factorise 3x² + 7x + 2.

Product = 3 × 2 = , Sum =

Numbers that work: and

= 3x² + 6x + x + 2 = 3x(x + 2) + 1(x + 2)

= ( + )(x + )

Excellent! 3x² + 7x + 2 = (3x + 1)(x + 2).

2. Factorise 6x² - 7x - 3.

Product = 6 × (-3) = , Sum =

Numbers: and

= 6x² - 9x + 2x - 3 = 3x(2x - 3) + 1(2x - 3)

= ( + )( - )

Perfect! 6x² - 7x - 3 = (3x + 1)(2x - 3).

3. Factorise x³ - 27.

Using a³ - b³ = (a - b)(a² + ab + b²)

Here, a = and b = (since 27 = 3³)

= (x - )(x² + x + )

Excellent! x³ - 27 = (x - 3)(x² + 3x + 9).

4. Factorise 8a³ + 27b³.

Using a³ + b³ = (a + b)(a² - ab + b²)

Here, 8a³ = (2a)³ and 27b³ = (3b)³

= ( + )(()² - ()() + ()²)

= (2a + 3b)(a² - ab + b²)

Perfect! 8a³ + 27b³ = (2a + 3b)(4a² - 6ab + 9b²).

5. Factorise 9x² - 25y².

Using a² - b² = (a + b)(a - b)

9x² = (3x)² and 25y² = (5y)²

= ( + )( - )

Great! 9x² - 25y² = (3x + 5y)(3x - 5y).

Part A: Section C – Long Answer Questions (4 Marks Each)

1. Factorise a³ + b³ + c³ - 3abc.

This is a special identity: a³ + b³ + c³ - 3abc = (a + b + c)(a² + b² + c² - ab - bc - caa² + b² + c² _ ab - bc + ca)

Formula verification:

When a + b + c = 0, then a³ + b³ + c³ = 3abc

So, a³ + b³ + c³ - 3abc = ()(a² + b² + c² - ab - bc - ca)

Perfect! This is one of the most important cubic identities.

2. Factorise x³ + 3x² - x - 3.

Group: (x³ + 3x²) + (-x - 3)

= (x + 3) - (x + 3)

= (x + )( - )

Now factorise (x² - 1) further:

= (x + 3)(x + )(x - )

Excellent! x³ + 3x² - x - 3 = (x + 3)(x + 1)(x - 1).

3. Factorise 2x³ + 3x² - 8x - 12 by grouping method.

Group: (2x³ + 3x²) + (-8x - 12)

= (2x + 3) - (2x + 3)

= (2x + )( - )

Now factorise (x² - 4):

= (2x + 3)(x + )(x - )

Perfect! 2x³ + 3x² - 8x - 12 = (2x + 3)(x + 2)(x - 2).

4. Factorise completely: x⁴ - y⁴.

First, use difference of squares: a² - b² = (a + b)(a - b)

x⁴ - y⁴ = (x²)² - (y²)² = ( + )(x² - y²)

Now factorise (x² - y²) further:

= (x² + y²)(x + )(x - )

Excellent! x⁴ - y⁴ = (x² + y²)(x + y)(x - y).

Note: (x² + y²) cannot be factorised further using real numbers.__

Part B: Objective Questions - Test Your Knowledge!

Answer these multiple choice questions:

6. a³ + b³ + c³ - 3abc =

(a) (a+b+c)(a²+b²+c²−ab−bc−ca)

(b) (a+b)(a²+b²)

(c) (a³+b³)(a+b)

(d) None

Perfect! This is a special cubic identity with three variables.

7. x² - 9x + 20 =

(a) (x - 5)(x - 4) (b) (x + 5)(x + 4) (c) (x - 10)(x - 2) (d) (x - 4)(x + 5)

Correct! Two numbers that multiply to 20 and add to -9 are -5 and -4.

8. 2x² - 3x - 9 =

(a) (2x + 3)(x - 3) (b) (2x - 3)(x + 3) (c) (x - 2)(2x + 3) (d) None

Excellent! Product = -18, sum = -3. Numbers: -6 and 3. Result: (2x + 3)(x - 3).

9. x² - 25 =

(a) (x - 5)(x + 5) (b) (x - 25)(x + 25) (c) (x + 5)² (d) None

Perfect! x² - 25 = x² - 5² = (x - 5)(x + 5).

10. (a + b)³ - (a - b)³ =

(a) 2b(3a² + b²) (b) 6ab² (c) 2a(3b² + a²) (d) None

Correct! Expand both cubes and simplify: = 6a²b + 2b³ = 2b(3a² + b²).

🎉 Exceptional Achievement! You've Mastered Advanced Factorisation!

Here's what you learned:

• Cubic Identities:

  • Sum of Cubes: a³ + b³ = (a + b)(a² - ab + b²)
  • Difference of Cubes: a³ - b³ = (a - b)(a² + ab + b²)
  • Examples: x³ - 27 = (x - 3)(x² + 3x + 9)
  • 8a³ + 27b³ = (2a + 3b)(4a² - 6ab + 9b²)

• Special Three-Variable Identity:

  • a³ + b³ + c³ - 3abc = (a + b + c)(a² + b² + c² - ab - bc - ca)
  • When a + b + c = 0, then a³ + b³ + c³ = 3abc

• Higher Power Factorisation:

  • x⁴ - y⁴ = (x² + y²)(x + y)(x - y)
  • Apply difference of squares repeatedly

• Complex Quadratics:

  • ax² + bx + c where a ≠ 1
  • Use splitting middle term method
  • Example: 6x² - 7x - 3 = (3x + 1)(2x - 3)

• Factorisation by Grouping:

  • For polynomials with 3 or more terms
  • Group terms strategically
  • Example: x³ + 3x² - x - 3 = (x + 3)(x + 1)(x - 1)

• Problem-Solving Strategy:

  1. Look for common factors first
  2. Identify special patterns (squares, cubes)
  3. Use appropriate identity
  4. Factor completely
  5. Verify by expansion

These advanced techniques are essential for solving higher-degree equations, calculus, and competitive examinations!