Question

If t2-1 is a factor of at3+t2-2t+b find a and b

Answer 1

Let p(t) = at3 + t2- 2t + b

Factorize t2- 1 as t2- 12 = (t-1)(t+1)

As t2- 1 is a factor of p(t)

Therefore, (t-1)(t+1) are also the factors of p(t)

p(1) = a(1)3+(1)2- 2(1)+ b

0 = a+ 1- 2+ b

0 = a- 1+ b

1 = a+ b

OR

a + b = 1 (1)

Also, p(-1) = a(-1)3+(-1)2- 2(-1)+ b

0 = -a+ 1+ 2+ b

3 = b - a

OR

b - a = 3 (2)

ADDING (1) and (2) -

a + b = 1

-a + b = 3

--------------

0 + 2b = 4

b = 4/2

b = 2

Substituting value of ' b ' in (1)

a + b = 1

a + 2 = 1

a = 1 - 2 = -1

Therefore , a = (-1)

b = 2