Step by Step Solution
Reformatting the input :
Changes made to your input should not affect the solution:
(1): "^-7" was replaced by "^(-7)".
Step by step solution :
Step 1 :
Equation at the end of step 1 :
(3x(-7) • x) - 7 = 0Step 2 :
Step 3 :
Pulling out like terms :
3.1 Pull out like factors :
3x(-6) - 7 = -x(-6) • (7x6 - 3)
Trying to factor as a Difference of Squares :
3.2 Factoring: 7x6 - 3
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 : 7 is not a square !!
Ruling : Binomial can not be factored as the
difference of two perfect squares
Polynomial Roots Calculator :
3.3 Find roots (zeroes) of : F(x) = 7x6 - 3
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 7 and the Trailing Constant is -3.
The factor(s) are:
of the Leading Coefficient : 1,7
of the Trailing Constant : 1 ,3
Let us test ....
| P | Q | P/Q | F(P/Q) | Divisor | |||||
|---|---|---|---|---|---|---|---|---|---|
| -1 | 1 | -1.00 | 4.00 | ||||||
| -1 | 7 | -0.14 | -3.00 | ||||||
| -3 | 1 | -3.00 | 5100.00 | ||||||
| -3 | 7 | -0.43 | -2.96 | ||||||
| 1 | 1 | 1.00 | 4.00 | ||||||
| 1 | 7 | 0.14 | -3.00 | ||||||
| 3 | 1 | 3.00 | 5100.00 | ||||||
| 3 | 7 | 0.43 | -2.96 |
Polynomial Roots Calculator found no rational roots
Equation at the end of step 3 :
-x(-6) • (7x6 - 3) = 0
Step 4 :
Theory - Roots of a product :
4.1 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 :
4.2 Solve : -x(-6) = 0
This equation has no solution !!
We actually looking at 1/
Solving a Single Variable Equation :
4.3 Solve : 7x6-3 = 0
Add 3 to both sides of the equation :
7x6 = 3
Divide both sides of the equation by 7:
x6 = 3/7 = 0.429
x = 6th root of (3/7)
The equation has two real solutions
These solutions are x = 6th root of ( 0.429) = ± 0.86830
Two solutions were found :
x = 6th root of ( 0.429) = ± 0.86830How did we do?
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