Question

I'll brainliest the answer ​

Answer 1

GIVEN:-

• P(x)=2x⁴-5x³+2x²-x+2

• f(x)= x²-3x+2

TO PROVE:-

• Without actual division we have to Prove the p(x) is divisible by f(x).

CONCEPT USED:-

• $\rm\pink{Factor\:theorem}$.

How To solve?

• First we have to find the factors or zeroes of p(x) by splitting the middle term if f(x).

• After finding the factors we have to use the factor theroem to to check if it is the zeroes of p(x) or not .

Now,

$\implies\rm{x^2-3x+2}$

$\implies\rm{x^2-2x-x+2}$

$\implies\rm{x(x-2)-1(x-2)}$

$\implies\rm{(x-1)(x-2)}$.

Hence, The factors of p(x) are (x-1) and (x-2).

Therefore,

$\sf{(x-1)=0}$

$\sf{x=1}$

Now, Put the value of x in given p(x).

$\implies\sf{P(1)=2(1)^4-5(1)^3+2(1)^2-1+2}$

$\implies\sf{2-5+2-1+2}$

$\implies\sf{-3+1+2}$

$\implies\sf{-3+3}$

$\implies\sf{0}$.

From this we get (x-1) is the factor of p(x).....1

Now, Similarly

$\sf{(x-2)=0}$

$\sf{x=2}$

Now, Put the value of x in given p(x).

$\implies\sf{P(1)=2(2)^4-5(2)^3+2(2)^2-2+2}$

$\implies\sf{32-40+8}$

$\implies\sf{-8+8}$

$\implies\sf{0}$

From this we get (x-2) is also the factor of p(x)...

From 1 and 2 it is proved that P(x) is exactly divisible by f(x).