Question

(x+2) is a factor of a polynomial ax3+bx2+x-6 and when this polynomial is divided by (x-2), it leaves the remainder 4. find the value of a and b

Answer 1

the ans is a=1/2 and b=1

Answer 2

Let p(x) = ax³ + bx² + x - 6 be the given polynomial.

Now,

( x + 2 ) is a factor of p(x).

Since, x + 2 = 0 ➠ x = '2

a( -2 )³ + b(-2)² + (-2) - 6 = 0 [ p(-2)=0 ]

➠ -8a + 4b - 2 - 6 = 0

➠ -8a + 4b = 8

➠ -2a + b = 2 ............. ( i )

It is given that p(x) leaves the remainder 4 when it divides by ( x - 2 ).

Since, x - 2 = 0 ➠ x = 2

a(2)³ + b(2)² + 2 - 6 = 4 [ p(2) = 4 ]

➠ 8a + 4b - 4 = 4

➠ 8a + 4b = 8

➠ 2a + b = 2 ..............( ii )

Adding ( i ) and ( ii ), we get

2b = 4 ➠ b = 2

Putting b = 2 in equation ( i ) , we get

-2a + 2 = 2 ➠ - 2a = 0 ➠ a = 0

Hence, a = 0 and b= 2