Solution - Nonlinear equations
Step by Step Solution
Reformatting the input :
Changes made to your input should not affect the solution:
(1): "p5" was replaced by "p^5".
Rearrange:
Rearrange the equation by subtracting what is to the right of the equal sign from both sides of the equation :
6*p^2-26*p^5-(p^2)=0
Step by step solution :
Step 1 :
Equation at the end of step 1 :
((6 • (p2)) - (2•13p5)) - p2 = 0Step 2 :
Equation at the end of step 2 :
((2•3p2) - (2•13p5)) - p2 = 0
Step 3 :
Step 4 :
Pulling out like terms :
4.1 Pull out like factors :
5p2 - 26p5 = -p2 • (26p3 - 5)
Trying to factor as a Difference of Cubes:
4.2 Factoring: 26p3 - 5
Theory : A difference of two perfect cubes, a3 - b3 can be factored into
(a-b) • (a2 +ab +b2)
Proof : (a-b)•(a2+ab+b2) =
a3+a2b+ab2-ba2-b2a-b3 =
a3+(a2b-ba2)+(ab2-b2a)-b3 =
a3+0+0-b3 =
a3-b3
Check : 26 is not a cube !!
Ruling : Binomial can not be factored as the difference of two perfect cubes
Polynomial Roots Calculator :
4.3 Find roots (zeroes) of : F(p) = 26p3 - 5
Polynomial Roots Calculator is a set of methods aimed at finding values of p for which F(p)=0
Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers p 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 26 and the Trailing Constant is -5.
The factor(s) are:
of the Leading Coefficient : 1,2 ,13 ,26
of the Trailing Constant : 1 ,5
Let us test ....
| P | Q | P/Q | F(P/Q) | Divisor | |||||
|---|---|---|---|---|---|---|---|---|---|
| -1 | 1 | -1.00 | -31.00 | ||||||
| -1 | 2 | -0.50 | -8.25 | ||||||
| -1 | 13 | -0.08 | -5.01 | ||||||
| -1 | 26 | -0.04 | -5.00 | ||||||
| -5 | 1 | -5.00 | -3255.00 | ||||||
| -5 | 2 | -2.50 | -411.25 | ||||||
| -5 | 13 | -0.38 | -6.48 | ||||||
| -5 | 26 | -0.19 | -5.18 | ||||||
| 1 | 1 | 1.00 | 21.00 | ||||||
| 1 | 2 | 0.50 | -1.75 | ||||||
| 1 | 13 | 0.08 | -4.99 | ||||||
| 1 | 26 | 0.04 | -5.00 | ||||||
| 5 | 1 | 5.00 | 3245.00 | ||||||
| 5 | 2 | 2.50 | 401.25 | ||||||
| 5 | 13 | 0.38 | -3.52 | ||||||
| 5 | 26 | 0.19 | -4.82 |
Polynomial Roots Calculator found no rational roots
Equation at the end of step 4 :
-p2 • (26p3 - 5) = 0
Step 5 :
Theory - Roots of a product :
5.1 A product of several terms equals zero.
When a product of two or more terms equals zero, then at least one of the terms must be zero.
We shall now solve each term = 0 separately
In other words, we are going to solve as many equations as there are terms in the product
Any solution of term = 0 solves product = 0 as well.
Solving a Single Variable Equation :
5.2 Solve : -p2 = 0
Multiply both sides of the equation by (-1) : p2 = 0
When two things are equal, their square roots are equal. Taking the square root of the two sides of the equation we get:
p = ± √ 0
Any root of zero is zero. This equation has one solution which is p = 0
Solving a Single Variable Equation :
5.3 Solve : 26p3-5 = 0
Add 5 to both sides of the equation :
26p3 = 5
Divide both sides of the equation by 26:
p3 = 5/26 = 0.192
When two things are equal, their cube roots are equal. Taking the cube root of the two sides of the equation we get:
p = ∛ 5/26
The equation has one real solution
This solution is p = ∛ 0.192 = 0.57721
Two solutions were found :
- p = ∛ 0.192 = 0.57721
- p = 0
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