Question

Consider a polynomial f(x)=ax^3+bx^+x+2÷3, if x+3 is a factor of f(x) and if f (x) is divided by x+ 2 then we get remainder as5 ​

Answer 1

Answer:

ANSWER

Let p(x)=ax

3

+bx

2

+x−6

Since (x+2) is a factor of p(x), then by Factor theorem p(−2)=0

⇒a(−2)

3

+b(−2)

2

+(−2)−6=0

⇒−8a+4b−8=0

⇒−2a+b=2 ...(i)

Also when p(x) is divided by (x-2) the remainder is 4, therefore by Remainder theorem p(2)=4

⇒a(2)

3

+b(2)

2

+2−6=4

⇒8a+4b+2−6=4

⇒8a+4b=8

⇒2a+b=2 ...(ii)

Adding equation (i) and (ii), we get

(−2a+b)+(2a+b)=2+2

⇒2b=4⇒b=2

Putting b=2 in (i), we get

−2a+2=2

⇒−2a=0⇒a=0

Hence, a=0 and b=2