Step by Step Solution
Reformatting the input :
Changes made to your input should not affect the solution:
(1): "1.7143" was replaced by "(17143/10000)".
Rearrange:
Rearrange the equation by subtracting what is to the right of the equal sign from both sides of the equation :
(1+x)^5-((17143/10000))=0
Step by step solution :
Step 1 :
17143
Simplify —————
10000
Equation at the end of step 1 :
17143
((x + 1)5) - ————— = 0
10000
Step 2 :
Rewriting the whole as an Equivalent Fraction :
2.1 Subtracting a fraction from a whole
Rewrite the whole as a fraction using 10000 as the denominator :
(x + 1)5 (x + 1)5 • 10000
(x + 1)5 = ———————— = ————————————————
1 10000
Equivalent fraction : The fraction thus generated looks different but has the same value as the whole
Common denominator : The equivalent fraction and the other fraction involved in the calculation share the same denominator
Adding fractions that have a common denominator :
2.2 Adding up the two equivalent fractions
Add the two equivalent fractions which now have a common denominator
Combine the numerators together, put the sum or difference over the common denominator then reduce to lowest terms if possible:
(x+1)5 • 10000 - (17143) 10000x5 + 50000x4 + 100000x3 + 100000x2 + 50000x - 7143
———————————————————————— = ———————————————————————————————————————————————————————
10000 10000
Trying to factor by pulling out :
2.3 Factoring: 10000x5 + 50000x4 + 100000x3 + 100000x2 + 50000x - 7143
Thoughtfully split the expression at hand into groups, each group having two terms :
Group 1: 100000x3 + 100000x2
Group 2: 50000x4 + 10000x5
Group 3: 50000x - 7143
Pull out from each group separately :
Group 1: (x + 1) • (100000x2)
Group 2: (x + 5) • (10000x4)
Group 3: (50000x - 7143) • (1)
Looking for common sub-expressions :
Group 1: (x + 1) • (100000x2)
Group 3: (50000x - 7143) • (1)
Group 2: (x + 5) • (10000x4)
Bad news !! Factoring by pulling out fails :
The groups have no common factor and can not be added up to form a multiplication.
Polynomial Roots Calculator :
2.4 Find roots (zeroes) of : F(x) = 10000x5 + 50000x4 + 100000x3 + 100000x2 + 50000x - 7143
Polynomial Roots Calculator is a set of methods aimed at finding values of x for which F(x)=0
Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers x which can be expressed as the quotient of two integers
The Rational Root Theorem states that if a polynomial zeroes for a rational number P/Q then P is a factor of the Trailing Constant and Q is a factor of the Leading Coefficient
In this case, the Leading Coefficient is 10000 and the Trailing Constant is -7143.
The factor(s) are:
of the Leading Coefficient : 1,2 ,4 ,5 ,8 ,10 ,16 ,20 ,25 ,40 , etc
of the Trailing Constant : 1 ,3 ,2381 ,7143
Let us test ....
| P | Q | P/Q | F(P/Q) | Divisor | |||||
|---|---|---|---|---|---|---|---|---|---|
| -1 | 1 | -1.00 | -17143.00 | ||||||
| -1 | 2 | -0.50 | -16830.50 | ||||||
| -1 | 4 | -0.25 | -14769.95 | ||||||
| -1 | 5 | -0.20 | -13866.20 | ||||||
| -1 | 8 | -0.12 | -12013.91 |
Note - For tidiness, printing of 75 checks which found no root was suppressed
Polynomial Roots Calculator found no rational roots
Equation at the end of step 2 :
10000x5 + 50000x4 + 100000x3 + 100000x2 + 50000x - 7143
——————————————————————————————————————————————————————— = 0
10000
Step 3 :
When a fraction equals zero :
3.1 When a fraction equals zero ...Where a fraction equals zero, its numerator, the part which is above the fraction line, must equal zero.
Now,to get rid of the denominator, Tiger multiplys both sides of the equation by the denominator.
Here's how:
10000x5+50000x4+100000x3+100000x2+50000x-7143
————————————————————————————————————————————— • 10000 = 0 • 10000
10000
Now, on the left hand side, the 10000 cancels out the denominator, while, on the right hand side, zero times anything is still zero.
The equation now takes the shape :
10000x5+50000x4+100000x3+100000x2+50000x-7143 = 0
Equations of order 5 or higher :
3.2 Solve 10000x5+50000x4+100000x3+100000x2+50000x-7143 = 0
Points regarding equations of degree five or higher.
(1) There is no general method (Formula) for solving polynomial equations of degree five or higher.
(2) By the Fundamental theorem of Algebra, if we allow complex numbers, an equation of degree n will have exactly n solutions
(This is if we count double solutions as 2 , triple solutions as 3 and so on
) (3) By the Abel-Ruffini theorem, the solutions can not always be presented in the conventional way using only a finite amount of additions, subtractions, multiplications, divisions or root extractions
(4) If F(x) is a polynomial of odd degree with real coefficients, then the equation F(X)=0 has at least one real solution.
(5) Using methods such as the Bisection Method, real solutions can be approximated to any desired degree of accuracy.
Approximating a root using the Bisection Method :
We now use the Bisection Method to approximate one of the solutions. The Bisection Method is an iterative procedure to approximate a root (Root is another name for a solution of an equation).
The function is F(x) = 10000x5 + 50000x4 + 100000x3 + 100000x2 + 50000x - 7143
At x= 0.00 F(x) is equal to -7143.00
At x= 1.00 F(x) is equal to 302857.00
Intuitively we feel, and justly so, that since F(x) is negative on one side of the interval, and positive on the other side then, somewhere inside this interval, F(x) is zero
Procedure :
(1) Find a point "Left" where F(Left) < 0
(2) Find a point 'Right' where F(Right) > 0
(3) Compute 'Middle' the middle point of the interval [Left,Right]
(4) Calculate Value = F(Middle)
(5) If Value is close enough to zero goto Step (7)
Else :
If Value < 0 then : Left <- Middle
If Value > 0 then : Right <- Middle
(6) Loop back to Step (3)
(7) Done!! The approximation found is Middle
Follow Middle movements to understand how it works :
Left Value(Left) Right Value(Right) 0.000000000 -7143.000 1.000000000 302857.00 0.000000000 -7143.000 1.000000000 302857.00 0.000000000 -7143.000 0.500000000 58794.50 0.000000000 -7143.000 0.250000000 13374.58 0.000000000 -7143.000 0.125000000 877.324707031 0.062500000 -3602.188 0.125000000 877.324707031 0.093750000 -1490.261 0.125000000 877.324707031 0.109375000 -339.804756299 0.125000000 877.324707031 0.109375000 -339.804756299 0.117187500 260.249204799 0.113281250 -41.883189364 0.117187500 260.249204799 0.113281250 -41.883189364 0.115234375 108.653881616 0.113281250 -41.883189364 0.114257812 33.253411948 0.113769531 -4.347828901 0.114257812 33.253411948 0.113769531 -4.347828901 0.114013672 14.444551059 0.113769531 -4.347828901 0.113891602 5.046301640 0.113769531 -4.347828901 0.113830566 0.348721595 0.113800049 -1.999682336 0.113830566 0.348721595 0.113815308 -0.825512543 0.113830566 0.348721595 0.113822937 -0.238403517 0.113830566 0.348721595 0.113822937 -0.238403517 0.113826752 0.055157028 0.113824844 -0.091623748 0.113826752 0.055157028 0.113825798 -0.018233486 0.113826752 0.055157028 0.113825798 -0.018233486 0.113826275 0.018461740 0.113825798 -0.018233486 0.113826036 0.000114119 0.113825917 -0.009059685 0.113826036 0.000114119 0.113825977 -0.004472783 0.113826036 0.000114119 0.113826007 -0.002179332 0.113826036 0.000114119 0.113826022 -0.001032607 0.113826036 0.000114119 0.113826029 -0.000459244 0.113826036 0.000114119 0.113826033 -0.000172562 0.113826036 0.000114119 0.113826035 -0.000029222 0.113826036 0.000114119 0.113826035 -0.000029222 0.113826036 0.000042449 0.113826035 -0.000029222 0.113826035 0.000006614 0.113826035 -0.000011304 0.113826035 0.000006614 0.113826035 -0.000002345 0.113826035 0.000006614
Next Middle will get us close enough to zero:
F( 0.113826035 ) is -0.000000105
The desired approximation of the solution is:
x ≓ 0.113826035
Note, ≓ is the approximation symbol
One solution was found :
x ≓ 0.113826035How did we do?
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