Question

if x^3+ax^2+bx+4 is divided by x-2 the remainder is 6. if it is divided by x+1 the remainder is -3 find a and b.​

Answer 1

Answer:

see below

Step-by-step explanation:

if x^3+ax^2+bx+4 is divided by x-2 the remainder is 6. if it is divided by x+1 the remainder is -3 find a and b.The value of a and b are -3 and -1 respectively.

Step-by-step explanation:

Given that (x-2) is a factor of polynomial

P(x)=x^3+ax^2+bx+6P(x)=x3+ax2+bx+6

Also when divided by (x - 3) leaves a remainder 3.

we have to find the value of a and b.

As 2 is the zero of the polynomial therefore by

P(2)=0

2^3+a(2)^2+2b+6=023+a(2)2+2b+6=0

8+4a+2b+6=08+4a+2b+6=0

4a+2b+14=04a+2b+14=0

2a+b=-72a+b=−7     →   (1)

\text{Also the polynomial }x^3+ax^2+bx+6\text{ when divided by (x - 3) leaves a remainder 3}Also the polynomial x3+ax2+bx+6 when divided by (x - 3) leaves a remainder 3

∴ P(3)=3P(3)=3

3^3+a(3)^2+3b+6=333+a(3)2+3b+6=3

27+9a+3b+6=327+9a+3b+6=3

9a+3b+33=39a+3b+33=3

3a+b=-103a+b=−10     →   (2)

Solving (1) and (2), we get

Subtracting equation (2) from (1)

2a+b-3a-b=-7-(-10)2a+b−3a−b=−7−(−10)

-a=3−a=3

a=-3a=−3

2a+b=-72a+b=−7

2(-3)+b=-72(−3)+b=−7

b=-7+6=-1b=−7+6=−1

Hence, the value of a and b are -3 and -1 respectively.