:
5
Simplify ββ
x2
Equation a
Step-by-step explanation: :
5
((-5x - ββ) + 8x) + 7
x2
STEP
2
:
Rewriting the whole as an Equivalent Fraction
2.1 Subtracting a fraction from a whole
Rewrite the whole as a fraction using x2 as the denominator :
-5x -5x β’ x2
-5x = βββ = ββββββββ
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:
-5x β’ x2 - (5) -5x3 - 5
ββββββββββββββ = ββββββββ
x2 x2
Equation at the end of step
2
:
(-5x3 - 5)
(ββββββββββ + 8x) + 7
x2
STEP
3
:
Rewriting the whole as an Equivalent Fraction :
3.1 Adding a whole to a fraction
Rewrite the whole as a fraction using x2 as the denominator :
8x 8x β’ x2
8x = ββ = βββββββ
1 x2
STEP
4
:
Pulling out like terms :
4.1 Pull out like factors :
-5x3 - 5 = -5 β’ (x3 + 1)
Trying to factor as a Sum of Cubes:
4.2 Factoring: x3 + 1
Theory : A sum of two perfect cubes, a3 + b3 can be factored into :
(a+b) β’ (a2-ab+b2)
Proof : (a+b) β’ (a2-ab+b2) =
a3-a2b+ab2+ba2-b2a+b3 =
a3+(a2b-ba2)+(ab2-b2a)+b3=
a3+0+0+b3=
a3+b3
Check : 1 is the cube of 1
Check : x3 is the cube of x1
Factorization is :
(x + 1) β’ (x2 - x + 1)
Trying to factor by splitting the middle term
4.3 Factoring x2 - x + 1
The first term is, x2 its coefficient is 1 .
The middle term is, -x its coefficient is -1 .
The last term, "the constant", is +1
Step-1 : Multiply the coefficient of the first term by the constant 1 β’ 1 = 1
Step-2 : Find two factors of 1 whose sum equals the coefficient of the middle term, which is -1 .
-1 + -1 = -2
1 + 1 = 2
Observation : No two such factors can be found !!
Conclusion : Trinomial can not be factored
Adding fractions that have a common denominator :
4.4 Adding up the two equivalent fractions
-5 β’ (x+1) β’ (x2-x+1) + 8x β’ x2 3x3 - 5
βββββββββββββββββββββββββββββββ = βββββββ
x2 x2
Equation at the end of step
4
:
(3x3 - 5)
βββββββββ + 7
x2
STEP
5
:
Rewriting the whole as an Equivalent Fraction :
5.1 Adding a whole to a fraction
Rewrite the whole as a fraction using x2 as the denominator :
7 7 β’ x2
7 = β = ββββββ
1 x2
Trying to factor as a Difference of Cubes:
5.2 Factoring: 3x3 - 5
Theory : A difference of two perfect cubes, a3 - b3 can be factored into
(a-b) β’ (a2 +ab +b2)
Proof : (a-b)β’(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 !!
Ruling : Binomial can not be factored as the difference of two perfect cubes
Polynomial Roots Calculator :
5.3 Find roots (zeroes) of : F(x) = 3x3 - 5
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 3 and the Trailing Constant is -5.
The factor(s) are:
of the Leading Coefficient : 1,3
of the Trailing Constant : 1 ,5
Let us test
P Q P/Q F(P/Q) Divisor
-1 1 -1.00 -8.00
-1 3 -0.33 -5.11
-5 1 -5.00 -380.00
-5 3 -1.67 -18.89
1 1 1.00 -2.00
1 3 0.33 -4.89
5 1 5.00 370.00
5 3 1.67 8.89
Polynomial Roots Calculator found no rational roots
Adding fractions that have a common denominator :
5.4 Adding up the two equivalent fractions
(3x3-5) + 7 β’ x2 3x3 + 7x2 - 5
ββββββββββββββββ = βββββββββββββ
x2 x2
Polynomial Roots Calculator :
5.5 Find roots (zeroes) of : F(x) = 3x3 + 7x2 - 5
See theory in step 5.3
In this case, the Leading Coefficient is 3 and the Trailing Constant is -5.
The factor(s) are:
of the Leading Coefficient : 1,3
of the Trailing Constant : 1 ,5
Let us test
P Q P/Q F(P/Q) Divisor
-1 1 -1.00 -1.00
-1 3 -0.33 -4.33
-5 1 -5.00 -205.00
-5 3 -1.67 0.56
1 1 1.00 5.00
1 3 0.33 -4.11
5 1 5.00 545.00
5 3 1.67 28.33
Polynomial Roots Calculator found no ratio
