Question

It is given a polynomial f(x) = 2x³ + ax² - bx + 12, where a and b are constants. When f(x) is divided by x-2, the remainder is -18. It is known that f(x) is divisible by x + 4. a) Find the values of a and b b) Solve the equation f(x) = 0​

Answer 1

Explanation:

(a). 2( 2 )³ + a ( 2 )² - b ( 2 ) + 12 + ( 18 ) = 0

16 + 4a - 2b + 30 = 0

4a - 2b = - 46

2a - b = - 23 ....... ( 1 )

2 ( - 4 )³ + a ( - 4 )² - b ( - 4 ) + 12 = 0

- 128 + 16a + 4b + 12 = 0

16a + 4b = 116

4a + b = 29 ........ ( 2 )

( 1 ) + ( 2 )

6a = 6 , a = 1

4(1) + b = 29 , b = 25

(b). 2x³ + x² + 25x + 12 = 0

`x_(1)` = - 4 ( given: f(x) is divisible by x + 4 )

`x_(2)` = 0.5

`x_(3)` = 3