Question

31) Without actual division, prove that x^4+2x^3-2x^2+2x-3 is exactly divisible by x^2+2x-3​

Answer 1

p(x) = x⁴ + 2x³ - 2x² + 2x - 3 = 0

First let's factorise the given divisor,  x² + 2x - 3​.

     x² + 2x - 3

⇒  x² + 3x - x - 3

⇒  x(x + 3) - (x + 3)

⇒  (x - 1)(x + 3)

So, if  p(x)  is divisible by  x² + 2x - 3,  then it will be divisible by both  x - 1  and  x + 3  each.

So we have to check whether  p(x)  is divisible by  x - 1  and  x + 3.

For this, without actual division, we can find it out by checking whether p(1) and p(-3) are equal to 0.

p(1) = (1)⁴ + 2(1)³ - 2(1)² + 2(1) - 3

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

p(1) = 0

p(-3) = (-3)⁴ + 2(-3)³ - 2(-3)² + 2(-3) - 3

p(-3) = 81 - 54 - 18 - 6 - 3

p(-3) = 0

So we got that  p(1) = p(-3) = 0.  Thus  x - 1  and  x + 3  are factors of p(x), thereby p(x) being divisible by  x² + 2x - 3​.

Hence Proved!