# Other Factorizations

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This solution deals with other factorizations.

Solution found

## Step by Step Solution

### Reformatting the input :

Changes made to your input should not affect the solution:

(1): "x6" was replaced by "x^6".

## Step 1 :

#### Equation at the end of step 1 :

` 40 - 5x`^{6}

## Step 2 :

## Step 3 :

#### Pulling out like terms :

3.1 Pull out like factors :

40 - 5x^{6} = -5 • (x^{6} - 8)

#### Trying to factor as a Difference of Squares :

3.2 Factoring: x^{6} - 8

Theory : A difference of two perfect squares, A^{2} - B^{2} can be factored into (A+B) • (A-B)

Proof : (A+B) • (A-B) =

A^{2} - AB + BA - B^{2} =

A^{2} - AB + AB - B^{2} =

A^{2} - B^{2}

Note : AB = BA is the commutative property of multiplication.

Note : - AB + AB equals zero and is therefore eliminated from the expression.

Check : 8 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) = x^{6} - 8

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 -8.

The factor(s) are:

of the Leading Coefficient : 1

of the Trailing Constant : 1 ,2 ,4 ,8

Let us test ....

P | Q | P/Q | F(P/Q) | Divisor | |||||
---|---|---|---|---|---|---|---|---|---|

-1 | 1 | -1.00 | -7.00 | ||||||

-2 | 1 | -2.00 | 56.00 | ||||||

-4 | 1 | -4.00 | 4088.00 | ||||||

-8 | 1 | -8.00 | 262136.00 | ||||||

1 | 1 | 1.00 | -7.00 | ||||||

2 | 1 | 2.00 | 56.00 | ||||||

4 | 1 | 4.00 | 4088.00 | ||||||

8 | 1 | 8.00 | 262136.00 |

Polynomial Roots Calculator found no rational roots

## Final result :

` -5 • (x`^{6} - 8)