## Typical Factorisation Solved Problems

Class 8/9/10 - Mathematics

Q1: Factorise x⁴ + x² + 1

Answer: x⁴ + x² + 1

= (x⁴ + 2x² + 1) - x²

Using identity (A + B)² = A² + B² + 2AB

= (x² + 1)² - x²

Using identity A² - B² = (A + B)(A - B)

= (x² + 1 + x)(x² + 1 - x) [Answer]

Q2: Factorise 25x² - 4y² - 9z² + 12yz

Answer: 25x² - 4y² - 9z² + 12yz

= 25x² - (4y² + 9z² - 12yz)

= (5x)² - [(2y)² + (3z)² + 2 × 2y × 3z]

= (5x)² - (2y - 3z)²

= (5x + 2y - 3z)(5x -2y + 3z) [Answer]

Q3: Factorise x⁸ - y⁸

= (x⁴)² - (y⁴)²

= (x⁴ + y⁴)(x⁴ - y⁴)

= (x⁴ + y⁴)[(x²)² - (y2)²]

= (x⁴ + y⁴)(x² + y²)(x² - y²)

= (x⁴ + y⁴)(x² + y²)(x + y)(x - y)

Q4: Factorise (y - x)² - 10(x - y) + 25

Answer: (y - x)² - 10(x - y) + 25

Let y - x = p and x - y = -p
∴ we have,
= p² + 10p + 25

= p² + 2(p)(5) + 5²

= (p + 5)²

= (y - x + 5)² [Answer]

Q5: Factorise 8a² - 22ab + 15b² using mid-term splitting

Answer: 8a² - 22ab + 15b²

Here Product = 8 × 15 = 120
Sum = -22

=  8a² - 10ab -12ab + 15b²

= 2a(4a - 5b) - 3b(4a - 5b)

= (4a - 5b)(2a - 3b)

Q6: Factorise 7√2x² - 10x -4√2

Here Product = 7√2 × -4√2 = -56
Sum = -10

= 7√2x² -14x + 4x -4√2

= 7√2x(x - √2) +4(x - √2)

= (x - √2)(7√2x + 4)