Question

Without actual division, prove that x4+2x3
-2x2+2x-3 is exactly divisible by x2+2x -3.

Answer 1

Answer:

$x^2+2x-3\text{ is factor of P(x)}$

Step-by-step explanation:

$\text{Let}P(x)=x^4+2x^3-2x^2+2x-3$

$x^2+2x-3=(x+3)(x-1)$

$\text{Now}$

$\text{put x= -3}$

$P(-3)=(-3)^4+2(-3)^3-2(-3)^2+2(-3)-3$

$P(-3)=81-54-18-6-3$

$P(-3)=0$

$\text{By factor theorem (x+3) is a factor of P(x)}$

$\text{since sum of the coefficients of P(x) is 0, (x-1) is a factor of P(x)}$

$\implies\text{(x+3) and (x-1) are factors of P(x)}$

$\implies\text{(x+3)(x-1) is factor of P(x)}$

$\implies\:x^2+2x-3\text{ is factor of P(x)}$