Solution - Factoring multivariable polynomials
Other Ways to Solve
Factoring multivariable polynomialsStep by Step Solution
Step by step solution :
Step 1 :
Step 2 :
Pulling out like terms :
2.1 Pull out like factors :
y - x2y = -y • (x2 - 1)
Trying to factor as a Difference of Squares :
2.2 Factoring: x2 - 1
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 the square of 1
Check : x2 is the square of x1
Factorization is : (x + 1) • (x - 1)
Equation at the end of step 2 :
(((21•((x•(y2))+x))•d)•x)+(-yd•(x+1)•(x-1)•y) = 0
Step 3 :
Multiplying exponential expressions :
3.1 y1 multiplied by y1 = y(1 + 1) = y2
Equation at the end of step 3 :
(((21•((x•(y2))+x))•d)•x)+-y2d•(x+1)•(x-1) = 0
Step 4 :
Step 5 :
Pulling out like terms :
5.1 Pull out like factors :
xy2 + x = x • (y2 + 1)
Polynomial Roots Calculator :
5.2 Find roots (zeroes) of : F(y) = y2 + 1
Polynomial Roots Calculator is a set of methods aimed at finding values of y for which F(y)=0
Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers y 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 1.
The factor(s) are:
of the Leading Coefficient : 1
of the Trailing Constant : 1
Let us test ....
| P | Q | P/Q | F(P/Q) | Divisor | |||||
|---|---|---|---|---|---|---|---|---|---|
| -1 | 1 | -1.00 | 2.00 | ||||||
| 1 | 1 | 1.00 | 2.00 |
Polynomial Roots Calculator found no rational roots
Equation at the end of step 5 :
((21x•(y2+1)•d)•x)+-y2d•(x+1)•(x-1) = 0
Step 6 :
Equation at the end of step 6 :
(21xd•(y2+1)•x)+-y2d•(x+1)•(x-1) = 0
Step 7 :
Multiplying exponential expressions :
7.1 x1 multiplied by x1 = x(1 + 1) = x2
Equation at the end of step 7 :
21x2d • (y2 + 1) + -y2d • (x + 1) • (x - 1) = 0
Step 8 :
Step 9 :
Pulling out like terms :
9.1 Pull out like factors :
20x2y2d + 21x2d + y2d = d • (20x2y2 + 21x2 + y2)
Trying to factor a multi variable polynomial :
9.2 Factoring 20x2y2 + 21x2 + y2
Try to factor this multi-variable trinomial using trial and error
Factorization fails
Equation at the end of step 9 :
d • (20x2y2 + 21x2 + y2) = 0
Step 10 :
Theory - Roots of a product :
10.1 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 :
10.2 Solve : d = 0
Solution is d = 0
Solving a Single Variable Equation :
10.3 Solve 20x2y2+21x2+y2 = 0
In this type of equations, having more than one variable (unknown), you have to specify for which variable you want the equation solved.
We shall not handle this type of equations at this time.
One solution was found :
d = 0How did we do?
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