Solution - Finding the roots of polynomials
Other Ways to Solve
Finding the roots of polynomialsStep by Step Solution
Step 1 :
Trying to factor as a Difference of Squares :
1.1 Factoring: x6-1000000
Theory : A difference of two perfect squares, A2 - B2 can be factored into (A+B) • (A-B)
Proof : (A+B) • (A-B) =
A2 - AB + BA - B2 =
A2 - AB + AB - B2 =
A2 - B2
Note : AB = BA is the commutative property of multiplication.
Note : - AB + AB equals zero and is therefore eliminated from the expression.
Check : 1 is not a square !!
Ruling : Binomial can not be factored as the
difference of two perfect squares
Polynomial Roots Calculator :
1.2 Find roots (zeroes) of : F(x) = x6-1000000
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 1 and the Trailing Constant is -1000000.
The factor(s) are:
of the Leading Coefficient : 1
of the Trailing Constant : 1 ,2 ,4 ,5 ,8 ,10 ,16 ,20 ,25 ,32 , etc
Let us test ....
| P | Q | P/Q | F(P/Q) | Divisor | |||||
|---|---|---|---|---|---|---|---|---|---|
| -1 | 1 | -1.00 | -999999.00 | ||||||
| -2 | 1 | -2.00 | -999936.00 | ||||||
| -4 | 1 | -4.00 | -995904.00 | ||||||
| -5 | 1 | -5.00 | -984375.00 | ||||||
| -8 | 1 | -8.00 | -737856.00 | ||||||
| -10 | 1 | -10.00 | 0.00 | x+10 | |||||
| -16 | 1 | -16.00 | 15777216.00 | ||||||
| -20 | 1 | -20.00 | 63000000.00 | ||||||
| -25 | 1 | -25.00 | 243140625.00 | ||||||
| -32 | 1 | -32.00 | 1072741824.00 | ||||||
| 1 | 1 | 1.00 | -999999.00 | ||||||
| 2 | 1 | 2.00 | -999936.00 | ||||||
| 4 | 1 | 4.00 | -995904.00 | ||||||
| 5 | 1 | 5.00 | -984375.00 | ||||||
| 8 | 1 | 8.00 | -737856.00 | ||||||
| 10 | 1 | 10.00 | 0.00 | x-10 | |||||
| 16 | 1 | 16.00 | 15777216.00 | ||||||
| 20 | 1 | 20.00 | 63000000.00 | ||||||
| 25 | 1 | 25.00 | 243140625.00 | ||||||
| 32 | 1 | 32.00 | 1072741824.00 |
The Factor Theorem states that if P/Q is root of a polynomial then this polynomial can be divided by q*x-p Note that q and p originate from P/Q reduced to its lowest terms
In our case this means that
x6-1000000
can be divided by 2 different polynomials,including by x-10
Polynomial Long Division :
1.3 Polynomial Long Division
Dividing : x6-1000000
("Dividend")
By : x-10 ("Divisor")
| dividend | x6 | - | 1000000 | ||||||||||||
| - divisor | * x5 | x6 | - | 10x5 | |||||||||||
| remainder | 10x5 | - | 1000000 | ||||||||||||
| - divisor | * 10x4 | 10x5 | - | 100x4 | |||||||||||
| remainder | 100x4 | - | 1000000 | ||||||||||||
| - divisor | * 100x3 | 100x4 | - | 1000x3 | |||||||||||
| remainder | 1000x3 | - | 1000000 | ||||||||||||
| - divisor | * 1000x2 | 1000x3 | - | 10000x2 | |||||||||||
| remainder | 10000x2 | - | 1000000 | ||||||||||||
| - divisor | * 10000x1 | 10000x2 | - | 100000x | |||||||||||
| remainder | 100000x | - | 1000000 | ||||||||||||
| - divisor | * 100000x0 | 100000x | - | 1000000 | |||||||||||
| remainder | 0 |
Quotient : x5+10x4+100x3+1000x2+10000x+100000 Remainder: 0
Polynomial Roots Calculator :
1.4 Find roots (zeroes) of : F(x) = x5+10x4+100x3+1000x2+10000x+100000
See theory in step 1.2
In this case, the Leading Coefficient is 1 and the Trailing Constant is 100000.
The factor(s) are:
of the Leading Coefficient : 1
of the Trailing Constant : 1 ,2 ,4 ,5 ,8 ,10 ,16 ,20 ,25 ,32 , etc
Let us test ....
| P | Q | P/Q | F(P/Q) | Divisor | |||||
|---|---|---|---|---|---|---|---|---|---|
| -1 | 1 | -1.00 | 90909.00 | ||||||
| -2 | 1 | -2.00 | 83328.00 | ||||||
| -4 | 1 | -4.00 | 71136.00 | ||||||
| -5 | 1 | -5.00 | 65625.00 | ||||||
| -8 | 1 | -8.00 | 40992.00 | ||||||
| -10 | 1 | -10.00 | 0.00 | x+10 | |||||
| -16 | 1 | -16.00 | -606816.00 | ||||||
| -20 | 1 | -20.00 | -2100000.00 | ||||||
| -25 | 1 | -25.00 | -6946875.00 | ||||||
| -32 | 1 | -32.00 | -25541472.00 | ||||||
| 1 | 1 | 1.00 | 111111.00 | ||||||
| 2 | 1 | 2.00 | 124992.00 | ||||||
| 4 | 1 | 4.00 | 165984.00 | ||||||
| 5 | 1 | 5.00 | 196875.00 | ||||||
| 8 | 1 | 8.00 | 368928.00 | ||||||
| 10 | 1 | 10.00 | 600000.00 | ||||||
| 16 | 1 | 16.00 | 2629536.00 | ||||||
| 20 | 1 | 20.00 | 6300000.00 | ||||||
| 25 | 1 | 25.00 | 16209375.00 | ||||||
| 32 | 1 | 32.00 | 48760992.00 |
The Factor Theorem states that if P/Q is root of a polynomial then this polynomial can be divided by q*x-p Note that q and p originate from P/Q reduced to its lowest terms
In our case this means that
x5+10x4+100x3+1000x2+10000x+100000
can be divided with x+10
Polynomial Long Division :
1.5 Polynomial Long Division
Dividing : x5+10x4+100x3+1000x2+10000x+100000
("Dividend")
By : x+10 ("Divisor")
| dividend | x5 | + | 10x4 | + | 100x3 | + | 1000x2 | + | 10000x | + | 100000 | ||
| - divisor | * x4 | x5 | + | 10x4 | |||||||||
| remainder | 100x3 | + | 1000x2 | + | 10000x | + | 100000 | ||||||
| - divisor | * 0x3 | ||||||||||||
| remainder | 100x3 | + | 1000x2 | + | 10000x | + | 100000 | ||||||
| - divisor | * 100x2 | 100x3 | + | 1000x2 | |||||||||
| remainder | 10000x | + | 100000 | ||||||||||
| - divisor | * 0x1 | ||||||||||||
| remainder | 10000x | + | 100000 | ||||||||||
| - divisor | * 10000x0 | 10000x | + | 100000 | |||||||||
| remainder | 0 |
Quotient : x4+100x2+10000 Remainder: 0
Trying to factor by splitting the middle term
1.6 Factoring x4+100x2+10000
The first term is, x4 its coefficient is 1 .
The middle term is, +100x2 its coefficient is 100 .
The last term, "the constant", is +10000
Step-1 : Multiply the coefficient of the first term by the constant 1 • 10000 = 10000
Step-2 : Find two factors of 10000 whose sum equals the coefficient of the middle term, which is 100 .
| -10000 | + | -1 | = | -10001 | ||
| -5000 | + | -2 | = | -5002 | ||
| -2500 | + | -4 | = | -2504 | ||
| -2000 | + | -5 | = | -2005 | ||
| -1250 | + | -8 | = | -1258 | ||
| -1000 | + | -10 | = | -1010 |
For tidiness, printing of 44 lines which failed to find two such factors, was suppressed
Observation : No two such factors can be found !!
Conclusion : Trinomial can not be factored
Final result :
(x4 + 100x2 + 10000) • (x + 10) • (x - 10)
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