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
![\left[\begin{array}{ccc}1&1&\\ \frac{1}{65} & \frac{1}{25} &\end{array}\right] \left[\begin{array}{ccc}x&\\y&\end{array}\right] = \left[\begin{array}{ccc}375&\\7&\end{array}\right]](https://tex.z-dn.net/?f=%20%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D1%261%26%5C%5C%20%5Cfrac%7B1%7D%7B65%7D%20%26%20%5Cfrac%7B1%7D%7B25%7D%20%26%5Cend%7Barray%7D%5Cright%5D%20%20%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7Dx%26%5C%5Cy%26%5Cend%7Barray%7D%5Cright%5D%20%3D%20%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D375%26%5C%5C7%26%5Cend%7Barray%7D%5Cright%5D%20)
![D= \left[\begin{array}{ccc}1&1\\ \frac{1}{65} & \frac{1}{25} \end{array}\right] =(1)( \frac{1}{25}) - (1)( \frac{1}{65} )= \frac{8}{325}](https://tex.z-dn.net/?f=D%3D%20%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D1%261%5C%5C%20%5Cfrac%7B1%7D%7B65%7D%20%26%20%5Cfrac%7B1%7D%7B25%7D%20%5Cend%7Barray%7D%5Cright%5D%20%3D%281%29%28%20%5Cfrac%7B1%7D%7B25%7D%29%20-%20%281%29%28%20%5Cfrac%7B1%7D%7B65%7D%20%29%3D%20%5Cfrac%7B8%7D%7B325%7D%20)
![D_{x} = \left[\begin{array}{ccc}375&1\\7& \frac{1}{25} \end{array}\right] = (375)( \frac{1}{25} )-(1)(7)=8](https://tex.z-dn.net/?f=%20D_%7Bx%7D%20%3D%20%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D375%261%5C%5C7%26%20%5Cfrac%7B1%7D%7B25%7D%20%5Cend%7Barray%7D%5Cright%5D%20%3D%20%28375%29%28%20%5Cfrac%7B1%7D%7B25%7D%20%29-%281%29%287%29%3D8)
