Question 3

If the zeroes of the polynomial x³– 3x²+ x + 1 are a – b, a, a + b, find a and b.

Answer

Given, p(x) = x³- 3x²+ x + 1

And zeroes are given as a – b, a, a + b

Now, comparing the given polynomial with general expression, we get;

∴ ax³+bx²+ cx + d = x³ – 3x²+ x + 1

a = 1, b = -3, c = 1 and d = 1

Sum of zeroes = a – b + a + a + b

-b/a = 3a

Putting the values b and a

- (-3)/1 = 3a

a = 1

Thus, the zeroes are 1 - b, 1, 1 + b.

Now, product of zeroes = 1(1 – b) (1 + b)

d/a = 1 - b²

-1/1 = 1- b²

b² = 1 + 1 = 2

b = √2

Hence, 1, -√2, 1, 1 + √2 are the zeroes of x³ – 3x²+ x + 1

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