Question

Q. The polynomials ax3 – 3x2 +4 and 2x3 – 5x +a when divided by (x – 2) leave the remainders p and q respectively. If p – 2q = 4, find the value of a.​

Answer 1

It is given that a polynomial ax  

3

−3x  

2

+4 when divided by (x−2) leaves the remainder p. Let us substitute x=2 in ax  

3

−3x  

2

+4 and equate it to p as follows:

a(2)  

3

−3(2)  

2

+4=p

⇒(a×8)−(3×4)+4=p

⇒8a−12+4=p

⇒8a−8=p....(1)

Similarly, given a polynomial 2x  

3

−5x+a when divided by (x−2) leaves the remainder q. Let us substitute x=2 in 2x  

3

−5x+a and equate it to q as follows:

2(2)  

3

−(5×2)+a=q

⇒(2×8)−10+a=q

⇒16−10+a=q

⇒6+a=q....(2)

It is also given that p−2q=4, we, now substitute the values of p and q from equations 1 and 2 as shown below:

p−2q=4

⇒8a−8−2(6+a)=4

⇒8a−8−12−2a=4

⇒6a−20=4

⇒6a=4+20

⇒6a=24

⇒a=  

6

24

​  

 

⇒a=4

Hence, a=4.