The polynomial ax3+bx2+x – 6 has (x+2) as a factor and leaves a remainder 4 when divided by (x–2). Find a and b.
0, 2
Let p(x)=ax3+bx2+x−6
By factor theorem, (x+2) can be a factor of p(x) only when p(−2)=0
p(−2)=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.
∴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 equations (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