Q. 194.6( 7 Votes )

# If both x+1 and x-1 are factors of *ax*^{3}+*x*^{2}-2*x*+*b*, find the value of *a* and *b*.

Answer :

Let, f (X) = *ax*^{3}+*x*^{2}-2*x*+*b* be the given polynomial.

Given (x + 1) and (x – 1) are factors of f (x).

From factor theorem,

If (x + 1) and (x – 1) are factors of f (x) then f (-1) = 0 and f (1) = 0 respectively.

f (-1) = 0

a (-1)^{3} + (-1)^{2} – 2 (-1) + b = 0

-a + 1 + 2 + b = 0

-a + 3 + b = 0

b – a + 3 = 0 (i)

f (1) = 0

a (1)^{3} + (1)^{2} – 2 (1) + b = 0

a + 1 – 2 + b = 0

a + b – 1 = 0

b + a – 1 = 0 (ii)

Adding (i) and (ii), we get

b – a + 3 + b + a – 1 = 0

2b + 2 = 0

2b = - 2

b = -1

Putting value of b in (i), we get

-1 - a + 3 = 0

-a + 2 = 0

a = 2

Hence, the value of a = 2 and b = -1.

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