Math Master

4x⁴ - x²
= x²(4x² - 1)
= x²{(2x)² - 1}
= x²(2x + 1)(2x - 1)

x² - y² + 4x + 4
= (x² + 4x + 4) - y²
= (x + 2)² - y²
= (x + 2 + y)(x + 2 - y)
= (x + y + 2)(x - y + 2)

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

4X² - 9X - 9
= 4x² - 12x + 3x - 9
= 4x(x - 3) + 3(x - 3)
= (x - 3)(4x + 3)

x² - y² - 2xz + z²
= x² - 2xz + z² - y²
= (x - z)² - y²
= (x + y - z)(x - y - z)

a⁴ + a²b² +b⁴
= (a²)² + 2.a².b² + (b²)² - a²b²
= (a² + b²)² - (ab)²
= (a² + ab + b²)(a² - ab + b²)

f(x) = x² - 4x + 3
∴ f(a) = a² - 4a + 3
f(a) = 0 হলে,
a² - 4a + 3 = 0
বা, a² - 3a - a + 3 = 0
বা, a(a - 3) - 1(a - 3) = 0
বা, (a - 3)(a - 1) = 0
∴ a = 3, 1

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

f(x) = 4x² + 5x - p,
x + 2 দ্বারা বিভাজ্য হলে f(-2) = 0
∴ 4(-2)² + 5(-2) - p = 0
বা, 16 - 10 - p = 0
বা, 6 - p = 0
∴ p = 6

যেহেতু f(x) = x³ - ax - 24 এর একটি উৎপাদক x - 3
∴ f(3) = 0
বা, 3³ - a.3 - 24 = 0
বা, 27 - 3a - 24 = 0
বা, 3a = 3
∴ a = 1

x³ - 6x² + 11x - 6
= x³ - x² - 5x² + 5x + 6x - 6
= x²(x - 1) - 5x(x - 1) + 6(x - 1)
= (x - 1)(x² - 5x + 6)
= (x - 1)(x - 2)(x - 3)

2a³ + 16
= 2(a³ + 8)
= 2(a³ + 2³)
= 2(a + 2)(a² - 2a + 4)

f(x) = x² + ax - 10 এর একটি উৎপাদক x - 5 হলে,
f(5) = 0
বা, 25 + 5a - 10 = 0
বা, 5a + 15 = 0
বা, 5a = -15
∴ a = -3

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

এখানে,
f(a) = a³ - 6a² + 16
∴ f(2) = 2³ - 6.2² + 16
= 8 - 24 + 16
= 24 - 24
= 0
∴ a - 2,
f(a) এর উৎপাদক

f(x) = 4x² + 11x + 6 = 0
এখানে,
f(-2) = 4.4 + 11(-2) + 6
= 16 - 22 + 6
= 0
∴ x = -2

f(x) = x² + 8x + 15 = 0
∴ f(-3) = (-3)² + 8(-3) + 15
= 9 - 24 + 15
= 24 - 24
= 0
∴ x = -3

f(x) = 2x³ + 5x² - 6x + 4
∴ f(-2) = 2(-2)³ + 5(-2)² - 6(-2) + 4
= - 16 + 20 + 12 + 4
= 20
অর্থাৎ ভাগশেষ 20,
∴ 20 বিয়োগ করতে হবে

যেহেতু,
f(5) = 0
বা, 5² + 7.5 + c = 0
বা, 25 + 35 + c = 0
বা, c + 60 = 0
∴ c = -60

x + 1 দ্বারা f(x) কে ভাগ করলে ভাগশেষ হবে
f(-1) = 7(-1)² - (-1) - 8
= 7 + 1 - 8
= 8 - 8
= 0