Math Master

a⁵ + 4a
= a(a⁴ + 4)
= a{(a²)² + 2²}
= a{(a² + 2)² - 2.a².2}
= a{(a² + 2)² - (2a)²}
= a(a² + 2a + 2)(a² - 2a + 2)

x⁴ - x²
= x²(x² - 1)
= x²(x + 1)(x - 1)

x² - y² - 2y - 1
= x² - (y² + 2y + 1)
= x² - (y + 1)²
= (x + y + 1)(x - y - 1)

x⁴ + x² + 1
= {(x²)² + 2.x².1 + 1²} - x²
= (x² + 1)² - x²
= (x² + x + 1)(x² - x + 1)

x² - 2bx + (a + b)(b - a)
= x² - 2bx + (b + a)(b - a)
= x² - 2bx + b² - a²
= (x - b)² - a²
= (x + a - b)(x - a - b)

x²+ 5x - 6
= x² + 6x - x - 6
= x(x + 6) -1(x + 6)
= (x +6)(x - 1)

x² + x - 2
= x² + 2x - x - 2
= x(x + 2) - 1(x + 2)
= (x + 2)(x - 1)

4x² - 13x - 12
= 4x² - 16x + 3x - 12
= 4x(x - 4) + 3(x - 4)
= (x - 4)(4x + 3)

2x² - xy - 6y²
= 2x² - 4xy + 3xy -6y²
= 2x(x - 2y) + 3y(x - 2y)
= (x - 2y)(2x + 3y)

x² + x - 240
= x² + 16x - 15x - 240
= x(x + 16) -15(x + 16)
= (x + 16)(x - 15)

x - 5, f(x) = x² + 7x + p এর উৎপাদক হলে,
f(5) = 0
বা, 25 + 35 + p = 0
বা, p + 60 = 0
∴ p = -60

এখানে, f(x) = x³ - 21x -20
∴ f(-1) = 0
∴ x + 1, f(x) এর একটি উৎপাদক

8x³ - 27
= (2x)³ - 3³
= (2x - 3){(2x)² + 2x.3 + 3²}
= (2x - 3)(4x² + 6x + 9)

x² + xy + yz - z²
= (x² - z²) + xy + yz
= (x + z)(x - z) + y(x + z)
= (x + z)(x + y - z)

এখানে, f(x) = x⁴ - 4x + 3
∴ f(1) = 1 - 4 + 3 = 0
∴ x - 1, f(x) এর একটি উৎপাদক

x - 3, f(x) = x³ + kx² - 6x - 9 এর উৎপাদক
∴ f(3) = 27 + 9k - 18 - 9 = 9k = 0
∴ k = 0

x⁶ - 1
= (x³)² - 1²
= (x³ + 1)(x³ - 1)
= (x + 1)(x² - x + 1)(x - 1)(x² + x + 1)

x² + 2ax - 2a - 1
= (x² - 1) + 2ax - 2a
= (x + 1)(x - 1) + 2a (x - 1)
= (x - 1)(x + 2a + 1)

4x⁴ + 1
= (2x²)² + 1
= (2x² + 1)² - 2.2x².1
= (2x² + 1)² - (2x)²
= (2x² + 2x + 1) (2x² - 2x + 1)