Question

if x2-1 is a factor of x3-ax2+bx+2 find the values of a and b and hence factorise the equation completly​

Answer 1

As we see here, x²-1=(x+1)(x-1) so it is divisible by both factors.

Solution: Let f(x)=x³-ax²+bx+2 then use factor theorem.

When we divide by x+1: The remainder is f(-1)

When we divide by x-1: The remainder is f(1)

Both remainders are 0. So we have f(-1)=0 and f(1)=0.

f(-1)=0 implies that: -1-a-b+2=0 ∴a+b=1

f(1)=0 implies that: 1-a+b+2=0 ∴a-b=3

The solutions are a=2 and b=-1.

Therefore, f(x)=x³-2x²-x+2 and is divisible by (x+1)(x-1).

Before we divide it, we know f(x) will have another solution α.

Solution: Let f(x)=(x+1)(x-1)(x-α) and compare the coefficients.

Needless to divide it actually, by comparison, the linear term is α=2. Therefore, the answer is (x+1)(x-1)(x-2).