if (x+1)(x+3)(x+5)(x+7) = 5760, find the real value of x

Answer:

+ 1 ) ( + 3 ) ( + 5 ) ( + 7 ) = 5760

x2 + 4x + 3 ) ( + 5 ) ( + 7 ) = 5760

x2 + 4x + 3 ) ( x2 + 12x + 35 ) = 5760 

x4 + 16x+ 86x2 + 176x  + 105 = 5760

x4 + 16x3 + 86x2 + 176x   = 5655

x4 + 16x3 + 86x2 + 176x  - 5655 = 0          ------------- ( 1 )

Now for real values of  we can check for factors of 5655 , 
And
5655 = 3 × 5 × 13 × 29
So our equation 1 must be satisfied by ( ±3 , ±5 , ±13 , ±29 )

Now we check for = -3 , we get our equation 1 , As:
(-3 )4 + 16( -3 )3 +86 ( -3 )2 + 176 ( -3 ) - 5655 = 0

81 - 432 + 774 - 528 - 5655 = 0

-5760  0 

So = -3 is not a solution of our equation.

Now check for = 3 We get

( 3)4 + 16( 3 )3 +86 ( 3 )2 + 176 ( 3 ) - 5655 = 0

81 + 432 + 774 + 528 - 5655 = 0
- 3840 ​  0 

So = 3 is not a solution of our equation.

Now check for = 5  , we get
( 5)4 + 16 ( 5 )3 + 86 ( 5 )2 + 176 ( 5 ) - 5655 = 0

625 + 2000 + 2150 + 880 - 5655 = 0

5655 - 5655 = 0
0 = 0 

SO = 5 is a solution of our given polynomial 

Now check for  = -5 , We get

625 - 2000 + 2150 - 880 - 5655 = 0

-5760  ​  0 
So = -5 is not a solution of our equation.

Now check for = 13 , we get

( 13 )4 + 16 ( 13 )3 + 86 ( 13 )2 + 176 ( 13 ) -5655 = 0

28561 + ​35152 + ​14534 + ​2288 - 5655 = 0
That is also not equal to zero 
So = 13 is not a solution of our equation.

Now for  -13 , we get

28561 - ​35152 + ​14534 - ​2288 - 5655 = 0

5655 - 5655 = 0

0 = 0 

So = -13 is a solution of our polynomial.

Now we check for = 29 ,
29 > 13 
And we get positive value as we put = 13 , So = 29 we get value even higher .
So,
= 29 is not a solution of our polynomial.

Now check for = -29
( -29 )4 + 16( -29 )3 + 86( -29 )2 + 176( -29 ) - 5655 = 0

707281  - ​390224  + ​72326  - ​5104 - 5655 = 0
That is also not equal to zero

So,
Our real value of  = 5  , -13                        ( Ans )

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