Solution - Finding the roots of polynomials
Other Ways to Solve
Finding the roots of polynomialsStep by Step Solution
Step 1 :
x
Simplify ——
x2
Dividing exponential expressions :
1.1 x1 divided by x2 = x(1 - 2) = x(-1) = 1/x1 = 1/x
Equation at the end of step 1 :
1 (((((x4)-(3•(x3)))-(13•(x2)))+(15•—))-6x)+5 xStep 2 :
Equation at the end of step 2 :
15 (((((x4)-(3•(x3)))-13x2)+——)-6x)+5 xStep 3 :
Equation at the end of step 3 :
15
(((((x4)-3x3)-13x2)+——)-6x)+5
x
Step 4 :
Rewriting the whole as an Equivalent Fraction :
4.1 Adding a fraction to a whole
Rewrite the whole as a fraction using x as the denominator :
x4 - 3x3 - 13x2 (x4 - 3x3 - 13x2) • x
x4 - 3x3 - 13x2 = ——————————————— = —————————————————————
1 x
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
Step 5 :
Pulling out like terms :
5.1 Pull out like factors :
x4 - 3x3 - 13x2 = x2 • (x2 - 3x - 13)
Trying to factor by splitting the middle term
5.2 Factoring x2 - 3x - 13
The first term is, x2 its coefficient is 1 .
The middle term is, -3x its coefficient is -3 .
The last term, "the constant", is -13
Step-1 : Multiply the coefficient of the first term by the constant 1 • -13 = -13
Step-2 : Find two factors of -13 whose sum equals the coefficient of the middle term, which is -3 .
| -13 | + | 1 | = | -12 | ||
| -1 | + | 13 | = | 12 |
Observation : No two such factors can be found !!
Conclusion : Trinomial can not be factored
Adding fractions that have a common denominator :
5.3 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:
x2 • (x2-3x-13) • x + 15 x5 - 3x4 - 13x3 + 15
———————————————————————— = ————————————————————
x x
Equation at the end of step 5 :
(x5 - 3x4 - 13x3 + 15)
(—————————————————————— - 6x) + 5
x
Step 6 :
Rewriting the whole as an Equivalent Fraction :
6.1 Subtracting a whole from a fraction
Rewrite the whole as a fraction using x as the denominator :
6x 6x • x
6x = —— = ——————
1 x
Checking for a perfect cube :
6.2 x5 - 3x4 - 13x3 + 15 is not a perfect cube
Trying to factor by pulling out :
6.3 Factoring: x5 - 3x4 - 13x3 + 15
Thoughtfully split the expression at hand into groups, each group having two terms :
Group 1: -13x3 + 15
Group 2: x5 - 3x4
Pull out from each group separately :
Group 1: (-13x3 + 15) • (1) = (13x3 - 15) • (-1)
Group 2: (x - 3) • (x4)
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 :
6.4 Find roots (zeroes) of : F(x) = x5 - 3x4 - 13x3 + 15
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 15.
The factor(s) are:
of the Leading Coefficient : 1
of the Trailing Constant : 1 ,3 ,5 ,15
Let us test ....
| P | Q | P/Q | F(P/Q) | Divisor | |||||
|---|---|---|---|---|---|---|---|---|---|
| -1 | 1 | -1.00 | 24.00 | ||||||
| -3 | 1 | -3.00 | -120.00 | ||||||
| -5 | 1 | -5.00 | -3360.00 | ||||||
| -15 | 1 | -15.00 | -867360.00 | ||||||
| 1 | 1 | 1.00 | 0.00 | x - 1 | |||||
| 3 | 1 | 3.00 | -336.00 | ||||||
| 5 | 1 | 5.00 | -360.00 | ||||||
| 15 | 1 | 15.00 | 563640.00 |
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 - 3x4 - 13x3 + 15
can be divided with x - 1
Polynomial Long Division :
6.5 Polynomial Long Division
Dividing : x5 - 3x4 - 13x3 + 15
("Dividend")
By : x - 1 ("Divisor")
| dividend | x5 | - | 3x4 | - | 13x3 | + | 15 | ||||||
| - divisor | * x4 | x5 | - | x4 | |||||||||
| remainder | - | 2x4 | - | 13x3 | + | 15 | |||||||
| - divisor | * -2x3 | - | 2x4 | + | 2x3 | ||||||||
| remainder | - | 15x3 | + | 15 | |||||||||
| - divisor | * -15x2 | - | 15x3 | + | 15x2 | ||||||||
| remainder | - | 15x2 | + | 15 | |||||||||
| - divisor | * -15x1 | - | 15x2 | + | 15x | ||||||||
| remainder | - | 15x | + | 15 | |||||||||
| - divisor | * -15x0 | - | 15x | + | 15 | ||||||||
| remainder | 0 |
Quotient : x4-2x3-15x2-15x-15 Remainder: 0
Polynomial Roots Calculator :
6.6 Find roots (zeroes) of : F(x) = x4-2x3-15x2-15x-15
See theory in step 6.4
In this case, the Leading Coefficient is 1 and the Trailing Constant is -15.
The factor(s) are:
of the Leading Coefficient : 1
of the Trailing Constant : 1 ,3 ,5 ,15
Let us test ....
| P | Q | P/Q | F(P/Q) | Divisor | |||||
|---|---|---|---|---|---|---|---|---|---|
| -1 | 1 | -1.00 | -12.00 | ||||||
| -3 | 1 | -3.00 | 30.00 | ||||||
| -5 | 1 | -5.00 | 560.00 | ||||||
| -15 | 1 | -15.00 | 54210.00 | ||||||
| 1 | 1 | 1.00 | -46.00 | ||||||
| 3 | 1 | 3.00 | -168.00 | ||||||
| 5 | 1 | 5.00 | -90.00 | ||||||
| 15 | 1 | 15.00 | 40260.00 |
Polynomial Roots Calculator found no rational roots
Adding fractions that have a common denominator :
6.7 Adding up the two equivalent fractions
(x4-2x3-15x2-15x-15) • (x-1) - (6x • x) x5 - 3x4 - 13x3 - 6x2 + 15
——————————————————————————————————————— = ——————————————————————————
x x
Equation at the end of step 6 :
(x5 - 3x4 - 13x3 - 6x2 + 15)
———————————————————————————— + 5
x
Step 7 :
Rewriting the whole as an Equivalent Fraction :
7.1 Adding a whole to a fraction
Rewrite the whole as a fraction using x as the denominator :
5 5 • x
5 = — = —————
1 x
Polynomial Roots Calculator :
7.2 Find roots (zeroes) of : F(x) = x5 - 3x4 - 13x3 - 6x2 + 15
See theory in step 6.4
In this case, the Leading Coefficient is 1 and the Trailing Constant is 15.
The factor(s) are:
of the Leading Coefficient : 1
of the Trailing Constant : 1 ,3 ,5 ,15
Let us test ....
| P | Q | P/Q | F(P/Q) | Divisor | |||||
|---|---|---|---|---|---|---|---|---|---|
| -1 | 1 | -1.00 | 18.00 | ||||||
| -3 | 1 | -3.00 | -174.00 | ||||||
| -5 | 1 | -5.00 | -3510.00 | ||||||
| -15 | 1 | -15.00 | -868710.00 | ||||||
| 1 | 1 | 1.00 | -6.00 | ||||||
| 3 | 1 | 3.00 | -390.00 | ||||||
| 5 | 1 | 5.00 | -510.00 | ||||||
| 15 | 1 | 15.00 | 562290.00 |
Polynomial Roots Calculator found no rational roots
Adding fractions that have a common denominator :
7.3 Adding up the two equivalent fractions
(x5-3x4-13x3-6x2+15) + 5 • x x5 - 3x4 - 13x3 - 6x2 + 5x + 15
———————————————————————————— = ———————————————————————————————
x x
Trying to factor by pulling out :
7.4 Factoring: x5 - 3x4 - 13x3 - 6x2 + 5x + 15
Thoughtfully split the expression at hand into groups, each group having two terms :
Group 1: -13x3 - 6x2
Group 2: x5 - 3x4
Group 3: 5x + 15
Pull out from each group separately :
Group 1: (13x + 6) • (-x2)
Group 2: (x - 3) • (x4)
Group 3: (x + 3) • (5)
Looking for common sub-expressions :
Group 1: (13x + 6) • (-x2)
Group 3: (x + 3) • (5)
Group 2: (x - 3) • (x4)
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 :
7.5 Find roots (zeroes) of : F(x) = x5 - 3x4 - 13x3 - 6x2 + 5x + 15
See theory in step 6.4
In this case, the Leading Coefficient is 1 and the Trailing Constant is 15.
The factor(s) are:
of the Leading Coefficient : 1
of the Trailing Constant : 1 ,3 ,5 ,15
Let us test ....
| P | Q | P/Q | F(P/Q) | Divisor | |||||
|---|---|---|---|---|---|---|---|---|---|
| -1 | 1 | -1.00 | 13.00 | ||||||
| -3 | 1 | -3.00 | -189.00 | ||||||
| -5 | 1 | -5.00 | -3535.00 | ||||||
| -15 | 1 | -15.00 | -868785.00 | ||||||
| 1 | 1 | 1.00 | -1.00 | ||||||
| 3 | 1 | 3.00 | -375.00 | ||||||
| 5 | 1 | 5.00 | -485.00 | ||||||
| 15 | 1 | 15.00 | 562365.00 |
Polynomial Roots Calculator found no rational roots
Final result :
x5 - 3x4 - 13x3 - 6x2 + 5x + 15
———————————————————————————————
x
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