23. Without actual division, prove that 2x* - 5x® + 2x? - x+2 is divi

1.	23. Without actual division, prove that 2x* - 5x® + 2x? - x+2 is divisible by2-3x+2.
Answer» Factorize: x²-3x+2x²-2x-1x+2x(x-2)-1(x-2)(x-2)(x-1)Therefore,(x-2)(x-1)are the factors. i) (x-2)x-2=0x=2So,p(x)=2p(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 . ii) (x-1)x-1=0x=1So,p(x)=1p(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=0Hence,it proves that (x-1) is a factor. Thus 2x⁴-5x³+2x²-x+2 is divisible by x²-3x+2

23. Without actual division, prove that 2x* - 5x® + 2x? - x+2 is divisible by2-3x+2.

Answer»

Factorize:

x²-3x+2x²-2x-1x+2x(x-2)-1(x-2)(x-2)(x-1)Therefore,(x-2)(x-1)are the factors.

i) (x-2)x-2=0x=2So,p(x)=2p(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 .

ii) (x-1)x-1=0x=1So,p(x)=1p(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=0Hence,it proves that (x-1) is a factor.

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