Question

What number should be added to the polynomial 2x3-3x2+x=0 so that it leaves remainder 3 when divided by x-2

Answer 1

The given polynomial is P(x)=2x

3

+3x

2

+ax+b

It is also given that if P(x) is divided by (x−2) then it will leave the remainder 2 and if divided by (x+2) then the remainder will be −2 which means that P(2)=2 and P(−2)=−2.

Let us first substitute P(2)=2 in P(x)=2x

3

+3x

2

+ax+b as shown below:

P(x)=2x

3

+3x

2

+ax+b

⇒P(2)=2(2)

3

+3(2)

2

+(a×2)+b

⇒2=(2×8)+(3×4)+2a+b

⇒2=16+12+2a+b

⇒2=28+2a+b

⇒2a+b=2−28

⇒2a+b=−26.........(1)

Now, substitute P(−2)=−2 in P(x)=2x

3

+3x

2

+ax+b as shown below:

P(x)=2x

3

+3x

2

+ax+b

⇒P(−2)=2(−2)

3

+3(−2)

2

+(a×−2)+b

⇒−2=(2×−8)+(3×4)−2a+b

⇒−2=−16+12−2a+b

⇒−2=−4−2a+b

⇒−2a+b=−2+4

⇒−2a+b=2.........(2)

Now adding the equations 1 and 2, we get

(2a−2a)+(b+b)=−26+2

⇒2b=−24

⇒b=−

2

24

⇒b=−12

Now substitute the value of b in equation 2:

−2a+(−12)=2

⇒−2a−12=2

⇒−2a=2+12

⇒−2a=14

⇒a=−

2

14

⇒a=−7

Hence a=−7 and b=−12

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