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