Question

what must be subtracted from x^3- 6x^2- 15x +80 so that the result is exactly divisible by x^2 +x -12

Answer 1

Let a should be subtracted,

Then,Remainder must be a,



                    ___________________
 x^2 + x -12 ) x^3 - 6x^2 - 15x + 80 (x - 7
                      x^3 +   x^2 - 12x 
                     -      -          + 
                     _________________
                              -7x^2 - 3x + 80
                              -7x^2 -7x +  84
                              +      +      -
                            _____________
                                      4x - 4
                                    ____________


Remainder = 4x - 4

4x - 4  should be subtracted.


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Method 2 : 

x^2 - x - 12 = x^2 - 4x + 3x - 12
                   = x(x - 4) + 3(x - 4)
                   = (x - 4)(x + 3) 



Let the number subtracted be k and the resulting polynomial be f(x) ,then 
f(x) = x^3 + 6x^2 - 15x + 80 - k 

Taking (x - 4),



Since x - 4 is a factor of f(x), by  factor theorem,

f(4) = 0 


⇒ x^3 - 6x^2 - 15x + 80 - k = 0 

⇒ (4)^3 - 6(4)^2 - 15(4) + 80 - k = 0 

⇒ 64 - 96 - 60 + 80 - k = 0 

⇒ -32 + 20 - k= 0 

⇒ -12 - k = 0 

⇒ k = -12 





=======================


Taking x + 3 



sinc x + 3 is a factor , by factor Theorem,

f(-3) =  0 

x^3 - 6x^2 - 15x + 80 - k = 0 

(-3)^3 - 6(-3) - 15(-3) + 80 - k =  0 

⇒ -27 + 18 + 45 + 80 - k = 0 

⇒ -9 + 125 - k = 0 

⇒ 116 - k= 0 

⇒ 116 = k 

=====================================


Then, 116 or -12 should be subtracted.




i hope this will help you



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