Get zeroes of a polynomial when some are known | example-1

Finding the Remaining Zeroes of the Polynomial

Given polynomial:
f(x) = ax⁴ - 4x³ + 4x - 1

Two given zeroes:
2 + √3 and 2 - √3

Step 1: Form a quadratic factor from the conjugate roots
(x - (2 + √3))(x - (2 - √3)) = [(x - 2) - √3][(x - 2) + √3]
= (x - 2)² - (√3)²
= x² - 4x + 1

So, (x² - 4x + 1) is a factor of f(x)

Assume a = 1:
f(x) = x⁴ - 4x³ + 4x - 1

Step 2: Perform polynomial division:
Perform the division of the polynomial (x⁴ - 4x³ + 0x² + 4x - 1) by the quadratic expression (x² - 4x + 1).

Step 3: Long Division

1) x⁴ ÷ x² = x²
   Multiply x² by the divisor (x² - 4x + 1), resulting in x⁴ - 4x³ + x².
   Subtract: (x⁴ - 4x³ + 0x²) - (x⁴ - 4x³ + x²) = -x²

2) Bring down 4x: -x² + 4x
   Divide: -x² ÷ x² = -1
   Multiply: -1(x² - 4x + 1) = -x² + 4x - 1
   Subtract: (-x² + 4x - 1) - (-x² + 4x - 1) = 0

No remainder ⇒ division is exact

So,
f(x) = (x² - 4x + 1)(x² - 1)

Step 4: Factor further:
x² - 1 = (x - 1)(x + 1)

Final factorization:
f(x) = (x - (2 + √3))(x - (2 - √3))(x - 1)(x + 1)

Other two zeroes are: x = 1 and x = -1