**Answer:**

**Step-by-step explanation:**

STEP

1

:

10

Simplify ——

x

Equation at the end of step

1

:

10

((((x4)-(5•(x2)))-6x)-——)-3

x

STEP

2

:

Equation at the end of step

2

:

10

((((x4) - 5x2) - 6x) - ——) - 3

x

STEP

3

:

Rewriting the whole as an Equivalent Fraction

3.1 Subtracting a fraction from a whole

Rewrite the whole as a fraction using x as the denominator :

x4 - 5x2 - 6x (x4 - 5x2 - 6x) • x

x4 - 5x2 - 6x = ————————————— = ———————————————————

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

4

:

Pulling out like terms

4.1 Pull out like factors :

x4 - 5x2 - 6x = x • (x3 - 5x - 6)

Polynomial Roots Calculator :

4.2 Find roots (zeroes) of : F(x) = x3 - 5x - 6

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 -6.

The factor(s) are:

of the Leading Coefficient : 1

of the Trailing Constant : 1 ,2 ,3 ,6

Let us test ....

P Q P/Q F(P/Q) Divisor

-1 1 -1.00 -2.00

-2 1 -2.00 -4.00

-3 1 -3.00 -18.00

-6 1 -6.00 -192.00

1 1 1.00 -10.00

2 1 2.00 -8.00

3 1 3.00 6.00

6 1 6.00 180.00

Polynomial Roots Calculator found no rational roots

Adding fractions that have a common denominator :

4.3 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:

x • (x3-5x-6) • x - (10) x5 - 5x3 - 6x2 - 10

———————————————————————— = ———————————————————

x x

Equation at the end of step

4

:

(x5 - 5x3 - 6x2 - 10)

————————————————————— - 3

x

STEP

5

:

Rewriting the whole as an Equivalent Fraction

5.1 Subtracting a whole from a fraction

Rewrite the whole as a fraction using x as the denominator :

3 3 • x

3 = — = —————

1 x

Checking for a perfect cube :

5.2 x5 - 5x3 - 6x2 - 10 is not a perfect cube

Trying to factor by pulling out :

5.3 Factoring: x5 - 5x3 - 6x2 - 10

Thoughtfully split the expression at hand into groups, each group having two terms :

Group 1: -6x2 - 10

Group 2: x5 - 5x3

Pull out from each group separately :

Group 1: (3x2 + 5) • (-2)

Group 2: (x2 - 5) • (x3)

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 :

5.4 Find roots (zeroes) of : F(x) = x5 - 5x3 - 6x2 - 10

See theory in step 4.2

In this case, the Leading Coefficient is 1 and the Trailing Constant is -10.

The factor(s) are:

of the Leading Coefficient : 1

of the Trailing Constant : 1 ,2 ,5 ,10

Let us test ....

P Q P/Q F(P/Q) Divisor

-1 1 -1.00 -12.00

-2 1 -2.00 -26.00

-5 1 -5.00 -2660.00

-10 1 -10.00 -95610.00

1 1 1.00 -20.00

2 1 2.00 -42.00

5 1 5.00 2340.00

10 1 10.00 94390.00

Polynomial Roots Calculator found no rational roots

Adding fractions that have a common denominator :

5.5 Adding up the two equivalent fractions

(x5-5x3-6x2-10) - (3 • x) x5 - 5x3 - 6x2 - 3x - 10

————————————————————————— = ————————————————————————

x x

Polynomial Roots Calculator :

5.6 Find roots (zeroes) of : F(x) = x5 - 5x3 - 6x2 - 3x - 10

See theory in step 4.2

In this case, the Leading Coefficient is 1 and the Trailing Constant is -10.

The factor(s) are:

of the Leading Coefficient : 1

of the Trailing Constant : 1 ,2 ,5 ,10

Let us test ....

P Q P/Q F(P/Q) Divisor

-1 1 -1.00 -9.00

-2 1 -2.00 -20.00

-5 1 -5.00 -2645.00

-10 1 -10.00 -95580.00

1 1 1.00 -23.00

2 1 2.00 -48.00

5 1 5.00 2325.00

10 1 10.00 94360.00

Polynomial Roots Calculator found no rational roots

Final result :

x5 - 5x3 - 6x2 - 3x - 10

————————————————————————

x