Question

I am confused in this can someone help?

Answer 1

`\huge\boxed{\sf a = -12 , b = 8}`

Explanation:

Given polynomial is:

`f(x) = ax^3+ b^2 + 3x + 4`

`\rule[225]{225}{2}`

First factor is (x-1)

Let x - 1 = 0 => x = 1

Putting this in the above polynomial

`f(1) = a(1)^3 + b(1)^2 + 3(1) + 4`

Given that it's remainder is 3

`3 = a + b + 3 + 4`

`3 = a + b + 7`

Subtracting 7 to both sides

`-4 = a + b\\OR \\`

a + b = -4 -------------------------- (1)

`\rule[225]{225}{2}`

Second Factor is 2x + 1

Let 2x + 1 = 0 => x = - 1 / 2

Putting in the above polynomial

`f(-(1)/(2) ) = a (-(1)/(2) )^3 + b (-(1)/(2) )^2 + 3 (-(1)/(2) ) + 4`

The remainder is 6

`6 = -(a)/(8) + (b)/(4) - (3)/(2) + 4\\6 - 4 = -(a)/(8) + (b)/(4) - (3)/(2)\\2 = (-a + 2b -12 )/(8) \\-a + 2b - 12 = 16\\-a + 2b = 16 + 12\\`

-a + 2b = 28 --------------------------------(2)

`\rule[225]{225}{2}`

Adding both equations

a + b - a + 2b = -4 + 28

3b = 24

Dividing both sides by 3

b = 8

`\rule[225]{225}{2}`

Putting b = 8 in Equation (1)

a + 8 = -4

a = - 4 - 8

a = -12

`\rule[225]{225}{2}`