Question

Without actual division prove that 2x⁴-5x³+2x³-x+2is divisible by x²-3x+2

Answer 1

x²-3x+2

=x²-2x-1x+2

=x(x-2)-1(x-2)

=(x-2)(x-1)

so that,,

(x-2)(x-1) are factors.

----> (x-2)

x-2=0

--> x=2

-->g(x)=2

g(x)=2x^4-5x^3+2x^2-x+2

g(2)=2(2)^4-5(2)^3+2(2)^2-2+2

=32-40+8

====>-40+40=0

so that (x-2) is a factor..

---> (x-1)

x-1=0

x=1

So,

g(x)=1

g(x)=2x^4-5x^3+2x^2-x+2

g(1)=2(1)^4-5(1)^3+2(1)^2-1+2

=2-5+2-1+2

=>>>6-6=0

(x-1) is also a factor!! :)