Question

what must be subtracted from 8the whole power 4 + 14 x cube minus 2 X square + 7 x minus 8 so that the resulting polynomial is exactly divisible by 4 x square + 3 x minus 2​

Answer 1

Answer:

14x - 10 should be subtracted

Step-by-step explanation:

what must be subtracted from 8x⁴ + 14x³ - 2x² + 7x - 8 so that the resulting polynomial is exactly divisible by 4x² + 3x - 2​

Let say resultant Polynomial  after subtracting = P(x)

P(x)  = (4x² + 3x - 2​)(ax² + bx + c)

=> P(x) = 4ax⁴ + (3a + 4b)x³ + (4c - 2a + 3b)x² + (3c - 2b)x  - 2c

4ax⁴ = 8x⁴ => a = 2

(3a + 4b)x³ = 14x³ => 3a + 4b = 14 => 6 + 4b = 14 => b = 2

(4c - 2a + 3b)x² =  - 2x² =>  4c - 4 + 6 = -2 => c = -1

(3c - 2b)x  = (-3 - 4)x = - 7x

- 2c = (-2)* (-1) = 2

P(x) = 8x⁴ + 14x³ - 2x² - 7x +2

Let say S is subtracted from 8x⁴ + 14x³ - 2x² + 7x - 8 to get P(x)

8x⁴ + 14x³ - 2x² + 7x - 8 - S  = 8x⁴ + 14x³ - 2x² - 7x +2

=>S = 14x - 10

14x - 10 should be subtracted