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
Answer: 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
Answer: 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)
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