(ii) Without actual division, prove that 2xin With2) Is a factor of th

1.	(ii) Without actual division, prove that 2xin With2) Is a factor of the polynomial -division, prove that 2x1-572 + 2x2 - x + 2 is divisible by x - 3x + 2.
Answer» x²-3x+2Sum=-3; product=2 x²-2x-1x+2x(x-2)-1(x-2)(x-2)(x-1)Therefore,(x-2)(x-1)are the factors. a)(x-2)x-2=0x=2So,p(x)=2 p(x)=2x⁴-5x³+2x²-x+2p(2)=2(2)⁴-5(2)³+2(2)²-2+2=32-40+8= -40+40=0Hence,it proves that (x-2) is a factor . b) (x-1)x-1=0x=1So,p(x)=1 p(x)=2x⁴-5x³+2x²-x+2p(1)=2(1)⁴-5(1)³+2(1)²-1+2=2-5+2-1+2=6-6=0 Hence,it proves that (x-1) is a factor. Thus 2x⁴-5x³+2x²-x+2 is divisible by x²-3x+2

(ii) Without actual division, prove that 2xin With2) Is a factor of the polynomial -division, prove that 2x1-572 + 2x2 - x + 2 is divisible by x - 3x + 2.

Answer»

x²-3x+2Sum=-3; product=2

x²-2x-1x+2x(x-2)-1(x-2)(x-2)(x-1)Therefore,(x-2)(x-1)are the factors.

a)(x-2)x-2=0x=2So,p(x)=2

p(x)=2x⁴-5x³+2x²-x+2p(2)=2(2)⁴-5(2)³+2(2)²-2+2=32-40+8= -40+40=0Hence,it proves that (x-2) is a factor .

b) (x-1)x-1=0x=1So,p(x)=1

p(x)=2x⁴-5x³+2x²-x+2p(1)=2(1)⁴-5(1)³+2(1)²-1+2=2-5+2-1+2=6-6=0

Hence,it proves that (x-1) is a factor.

Thus 2x⁴-5x³+2x²-x+2 is divisible by x²-3x+2