-2 • (2x4 - 15x3 - 54x2 + 4x + 21)
——————————————————————————————————
x2
Step by step solution :
Step 1 :
2
Simplify ——
x2
Equation at the end of step 1 :
18 2
(6x-————)-((4•(x+3))•((x+——)-9))
(x2) x2
Step 2 :
Rewriting the whole as an Equivalent Fraction :
2.1 Adding a fraction to a whole
Rewrite the whole as a fraction using x2 as the denominator :
x x • x2
x = — = ——————
1 x2
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
Adding fractions that have a common denominator :
2.2 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 • x2 + 2 x3 + 2
—————————— = ——————
x2 x2
Equation at the end of step 2 :
18 (x3+2)
(6x-————)-((4•(x+3))•(——————-9))
(x2) x2
Step 3 :
Rewriting the whole as an Equivalent Fraction :
3.1 Subtracting a whole from a fraction
Rewrite the whole as a fraction using x2 as the denominator :
9 9 • x2
9 = — = ——————
1 x2
Trying to factor as a Sum of Cubes :
3.2 Factoring: x3 + 2
Theory : A sum of two perfect cubes, a3 + b3 can be factored into :
(a+b) • (a2-ab+b2)
Proof : (a+b) • (a2-ab+b2) = a3-a2b+ab2+ba2-b2a+b3 =
a3+(a2b-ba2)+(ab2-b2a)+b3=
a3+0+0+b3=
a3+b3
Check : 2 is not a cube !! Ruling : Binomial can not be factored as the difference of two perfect cubes
3.3 Find roots (zeroes) of : F(x) = x3 + 2
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 2.
The factor(s) are:
of the Leading Coefficient : 1
of the Trailing Constant : 1 ,2
Let us test ....
P Q P/Q F(P/Q) Divisor
-1 1 -1.00 1.00 -2 1 -2.00 -6.00 1 1 1.00 3.00 2 1 2.00 10.00
Adding fractions that have a common denominator :
3.4 Adding up the two equivalent fractions
(x3+2) - (9 • x2) x3 - 9x2 + 2
————————————————— = ————————————
x2 x2 Equation at the end of step 3 :
18 (x3-9x2+2)
(6x-————)-((4•(x+3))•——————————)
(x2) x2
Step 4 :
Equation at the end of step 4 :
18 (x3-9x2+2)
(6x-————)-(4•(x+3)•——————————)
(x2) x2
Step 5 :
5.1 Find roots (zeroes) of : F(x) = x3-9x2+2
See theory in step 3.3 In this case, the Leading Coefficient is 1 and the Trailing Constant is 2.
The factor(s) are:
of the Leading Coefficient : 1
of the Trailing Constant : 1 ,2
Let us test ....
P Q P/Q F(P/Q) Divisor
-1 1 -1.00 -8.00 -2 1 -2.00 -42.00 1 1 1.00 -6.00 2 1 2.00 -26.00
Equation at the end of step 5 :
18 4•(x+3)•(x3-9x2+2)
(6x-————)-——————————————————
(x2) x2
Step 6 :
18
Simplify ——
x2
Equation at the end of step 6 :
18 4 • (x + 3) • (x3 - 9x2 + 2)
(6x - ——) - ————————————————————————————
x2 x2 Step 7 :
Rewriting the whole as an Equivalent Fraction :
7.1 Subtracting a fraction from a whole
Rewrite the whole as a fraction using x2 as the denominator :
6x 6x • x2
6x = —— = ———————
1 x2 Adding fractions that have a common denominator :
7.2 Adding up the two equivalent fractions
6x • x2 - (18) 6x3 - 18
—————————————— = ————————
x2 x2 Equation at the end of step 7 :
(6x3 - 18) 4 • (x + 3) • (x3 - 9x2 + 2)
—————————— - ————————————————————————————
x2 x2 Step 8 :
Step 9 :
Pulling out like terms :
9.1 Pull out like factors :
6x3 - 18 = 6 • (x3 - 3)
Trying to factor as a Difference of Cubes:
9.2 Factoring: x3 - 3
Theory : A difference of two perfect cubes, a3 - b3 can be factored into
(a-b) • (a2 +ab +b2)
Proof : (a-b)•(a2+ab+b2) =
a3+a2b+ab2-ba2-b2a-b3 =
a3+(a2b-ba2)+(ab2-b2a)-b3 =
a3+0+0+b3 =
a3+b3
Check : 3 is not a cube !!
9.3 Find roots (zeroes) of : F(x) = x3 - 3
See theory in step 3.3 In this case, the Leading Coefficient is 1 and the Trailing Constant is -3.
The factor(s) are:
of the Leading Coefficient : 1
of the Trailing Constant : 1 ,3
Let us test ....
P Q P/Q F(P/Q) Divisor
-1 1 -1.00 -4.00 -3 1 -3.00 -30.00 1 1 1.00 -2.00 3 1 3.00 24.00
Adding fractions which have a common denominator :
9.4 Adding fractions which have a common denominator
Combine the numerators together, put the sum or difference over the common denominator then reduce to lowest terms if possible:
6 • (x3-3) - (4 • (x+3) • (x3-9x2+2)) -4x4 + 30x3 + 108x2 - 8x - 42
————————————————————————————————————— = —————————————————————————————
x2 x2
Step 10 :
Pulling out like terms :
10.1 Pull out like factors :
-4x4 + 30x3 + 108x2 - 8x - 42 =
-2 • (2x4 - 15x3 - 54x2 + 4x + 21)