Question

what must be added to x cube minus 6 x square - 15 x + 80 so that the result is exactly divisible by x^2 + X - 12?

Answer 1

Thus, -4x+4 should be added to the given polynomial. Using Remainder theorem.

Answer 2

Answer :

• x³ - 6x² - 15x + 80 will be divisible by x² + x - 12, if 4x - 4 is subtracted from it.

Step-by-step explanation

Let p(x) = x³ - 6x² - 15x + 80 and q(x) = x² + x - 12. When p(x) is divided by q(x), the remainder is a linear expression in x.

So, let r(x) =ax + b is subtracted from p(x) so that p(x) - r(x) is divisible by q(x).

Let f ( x ) = p ( x ) - r ( x ) . Then ,

f(x ) = x³ - 6x² - 15x + 80 - ax - b

= x³ + 6x² - x ( 15 + a ) + 80 - b

We have q ( x ) = x² + x - 12 = x² + 4x - 3x - 12

= x ( x + 4 ) - 3 ( x + 4 ) = ( x - 3 ) ( x + 4 )

Clearly , ( x ) is divisible by ( x - 3 ) and ( x + 4 ) i . e . , ( x - 3 ) and ( x + 4 ) are factors of q ( x ) .

Therefore , f(x) will be divisible by q( x ) , if ( x - 3 ) and ( x + 4 ) are factors of f ( x ) i . e ., f ( 3 ) = 0 and f ( - 4 ) = 0

➟ 3³ - 6 × 3² - 3 ( 15 + a ) + 80 - b = 0

and( - 4 )³ - 6 ( - 4 )² - (- 4 ) x ( 15 + a ) + 80 - b =0

➟ 27 - 54 - 45 - 3a + 80 - b = 0 and - 64 - 96 + 60 + 4a + 80 - b = 0

➟ 8 - 3a - b = 0 and - 20 + 4a - b = 0

➟ 3a + b = 8 and 4a - b = 20

Adding these two equations , we get

3a + b + 4a - b = 8 + 20

➟ 7a = 28

➟ a = 4

Putting a = 4 in 3a + b = 8 , we get

3 x 4 + b = 8

➟ b = - 4

∴ r ( x ) = ax + b = 4x - 4

Hence, x³ - 6x² - 15x + 80 will be divisible by x² + x - 12, if 4x - 4 is subtracted from it.