Question

If x - √5 is a factor of the cubic polynomial x³ - 35x² + 13x - 35, then find all the zeroes of the polynomial.

Answer 1

Let P(x) = x³ - 35x² + 13x - 35

As √5 is one of the zeroes of P(x).

⇒ (x - √5) is one of the factors of P(x).

Dividend = (divisor) (quotient) + remainder

⇒ p(x) = g(x).q(x) + r(x)

⇒ x³ - 35x² + 13x - 35 = (x - √5) (x² - 2√5x + 3) + 0

x³ - 35x² + 13x - 35 = 0

= (x - √5) (x² - 2√5x + 3) = 0

= (x - √5) [x² - {(√5 + √2) x + (√5 - √2) x} + 3) = 0

= (x - √5) [x{x - (√5 + √2)} - (√5 - √2) {x - (√5 + √2)} + 3] = 0

= (x - √5) {x - (√5 + √2)} {x - (√5 - √2)} = 0

⇒ x = (√5 + √2), (√5 - √2), √5