Question

Given that the zeroes of the cubic polynomial x x x 3 2 − + + 6 3 10 are of form a a b , + , a b + 2 for some real numbers a and b, find the values of a and b as well as the zeroes of the given polynomial.

Answer 1

• a = 5, b = - 3

• a = - 1, b = 3

• Zeroes are 5, 2, - 1

Complete question: Given that the zeroes of the cubic polynomial x3 - 6x2 + 3x + 10 are of the form a, a + b, a + 2b for some real numbers a and b, find the values of a and b as well as the zeroes of the given polynomial.

Step-by-step explanation:

The given polynomial is

f(x) = x3 - 6x2 + 3x + 10

Since a, a + b, a + 2b are the zeroes of f(x), by the relation between zeroes and coefficients, we get

a + (a + b) + (a + 2b) = - (- 6)/1

or, a + a + b + a + 2b = 6

or, 3a + 3b = 6

or, a + b = 2 ..... (1)

and a (a + b) (a + 2b) = - 10/1

or, a (a + 2 - a) (a + 4 - 2a) = - 10 [ by (1) ]

or, a (2) (4 - a) = - 10

or, a (4 - a) = - 5

or, a2 - 4a - 5 = 0

or, (a - 5) (a + 1) = 0

Either a - 5 = 0 or, a + 1 = 0

So a = 5 or, a = - 1

• When a = 5, b = - 3

• When a = - 1, b = 3

For a = 5, b = - 3, the zeroes are 5, 2, - 1

For a = - 1, b = 3, the zeroes are - 1, 2, 5

∴ the zeroes of the given polynomial are 5, 2, - 1

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Answer 2

Explanation:

Factorise:

(i)

3 2

x x x    2 2