Question

without actual division prove that 2x^4-5x^3+2x^2-x+2 is exactly divisible by x^2-3x+2

Answer 1

x^2 -3x+2
x^2-2x-1x+2
x(x-2)-1(x-2)
(x-2)(x-1)
f(2)=2(2)^4-5(2)^3+2(2)^2-2+2
32-40+8-2+2=0
f(1)=2(1)^4-5(1)^3+2(1)^2-1+2=2-5+2-1+2= 0
0+0= 0

Answer 2

GIVEN
g(x) =x^2-3x+2 ............................. (1)
p(x) =2x^4-5x^3+2x^2-x+2 .......... (2)

Factorize and find out the value of x of (1)
g(x) =x^2-3x+2
x^2 - 2x - x + 2
x (x - 2) - 1 (x - 2)
(x - 2) (x - 1)
Now make this equal to zero for finding x value
(x - 2) (x - 1) = 0
x - 2 = 0 & x - 1 = 0
x = 2 & x = 1

Now replace the value of x in (2) it should be equal to zero.
p(x) =2x^4-5x^3+2x^2-x+2

Value x = 2
2(2)^4 - 5(2)^3 + 2(2)^2 - (2) + 2
= (2 x 16) - (5 x 8) + (2 x 4) - 2 + 2
= 32 - 40 + 8 - 2 + 2
= 42 - 42
= 0

Value x = 1
2(1)^4 - 5(1)^3 + 2(1)^2 - (1) + 2
= (2 x 1) - (5 x 1) + (2 x 1) - 1 + 2
= 2 - 5 + 2 - 1 + 2
= 6 - 6
= 0

Both values of x of g(x) when replenish in p(x) we get zero as result. Therefore, p(x) is fully divisible by g(x).