Question

4. Without actual division, prove that 24 - 5x + 2x - + 2 is divisible by - 2
Hint: Factorise x - 3x + 2)​

Answer 1

Question:-

Without actual division, prove that 24 - 5x + 2x - + 2 is divisible by - 2

Solution :-

First of all,

★ x^2-3x+2

⇒x^2-2x-1x+2

⇒x(x-2)-1(x-2)

⇒(x-2)(x-1)

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

★ (x-2)

⇒x-2=0

⇒x=2

• So,p(x)=2

➺ p(x)=2x^4-5x^3+2x^2-x+2

⇒p(2)=2(2)^4-5(2)^3+2(2)^2-2+2

⇒32-40+8

⇒ -40+40=0

• Hence,it proves that (x-2) is a factor .

★ (x-1)

⇒x-1=0

⇒x=1

• So,p(x)=1

➺ p(x)=2x^4-5x^3+2x^2-x+2

⇒p(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.