Question

The polynomial px=ax3-3x2+4 and gx=2x3-5x+a when divided by (x-2) and (x-3) leave the remainder p and q respectively. Find the value of a if p-2q=4.​

Answer 1

Answer:

The polynomial px=ax3-3x2+4 and gx=2x3-5x+a when divided by (x-2) and (x-3) leave the remainder p and q respectively. Find the value of a if p-2q=4

Answer 2

Answer:

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.