If 4 is a zero of the cubic polynomial x 3 - 3 x 2 - 10 x + 24 , find its other two zeroes.

Let f ( x ) = x 3 - 3 x 2 - 10 x + 24 and given that, 4 is one of the zeroes of f ( x ) i.e., ( x − 4 ) is one of the factor of given cubic polynomial. Now, using division algorithm, So, the other factor of the polynomial is x 2 + x - 6 . From middle term factorization, we have x 2 + x - 6 = x 2 + 3 x - 2 x - 6 = x ( x + 3 ) - 2 ( x + 3 ) = x - 2 x + 3 ⇒ x = 2 , x = - 3 Hence, other two zeroes are 2 , - 3 .

Divide the polynomial f ( x ) = 3 x 2 - x 3 - 3 x + 5 by the polynomial g ( x ) x - 1 - x 2 and verify the division algorithm

The given polynomials are not in standard form. To carry out division, we first write both the dividend and divisor in decreasing orders of their degrees. So, dividend - x 3 + 3 x 2 - 3 x + 5 and divisor is - x 2 + x - 1 So, quotient = x - 2 , remainder = 3 Divisor × Quotient + Remainder = Dividend - x 2 + x - 1 × x - 2 + 3 = - x 3 + 3 x 2 - 3 x + 5 - x 3 + x 2 - x + 2 x 2 - 2 x + 2 + 3 = - x 3 + 3 x 2 - 3 x + 5 - x 3 + 3 x 2 - 3 x + 5 = - x 3 + 3 x 2 - 3 x + 5 Thus, the division algorithm is verified

Find all zeroes of the polynomial 2 x 4 - 9 x 3 + 5 x 2 + 3 x - 1 if two of its zeroes are 2 + 3 and 2 - 3 .

Given, 2 x 4 - 9 x 3 + 5 x 2 + 3 x - 1 And ( 2 + 3 ) and ( 2 - 3 ) are the zeroes of the given polynomial. Product of the zeroes = x - 2 + 3 x - 2 - 3 = x 2 - x 2 - 3 - x 2 + 3 + 2 2 - 3 2 = x 2 - 2 x + x 3 - 2 x - x 3 + 4 - 3 = x 2 - 4 x + 1 Product of the zeroes of the given polynomial is a factor of the polynomial. So, 2 x 4 - 9 x 3 + 5 x 2 + 3 x - 1 is divisible by x 2 - 4 x + 1 . By using long division: 2 x 2 - x - 1 x 2 - 4 x + 1 2 x 4 - 9 x 3 + 5 x 2 + 3 x - 1 2 x 4 - 8 x 3 + 2 x 2 - + - - x 3 + 3 x 2 + 3 x - 1 - x 3 + 4 x 2 - x + - + - x 2 + 4 x - 1 - x 2 + 4 x - 1 + - + 0 ∴ 2 x 4 - 9 x 3 + 5 x 2 + 3 x - 1 = x 2 - 4 x + 1 2 x 2 - x - 1 = x 2 - 4 x + 1 2 x 2 - 2 x + x - 1 = x 2 - 4 x + 1 2 x ( x - 1 ) + 1 ( x - 1 ) = x 2 - 4 x + 1 ( x - 1 ) ( 2 x + 1 ) Here, ( x - 1 ) ( 2 x + 1 ) = 0 ⇒ x - 1 = 0 or 2 x + 1 = 0 ⇒ x = 1 or ⇒ x = - 1 2 Hence, all the zeroes of the polynomial are 1 , - 1 2 , 2 + 3 and 2 - 3 .