simplify 5t3 / 4t-8*6t-12/10t
Answers
Answer:
Step-by-step explanation:
t • (25t3 - 984)
————————————————
20
Step by step solution :
Step 1 :
6
Simplify —
5
Equation at the end of step 1 :
(t3) 6
(((5•————)•t)-48t)-(—•t)
4 5
Step 2 :
t3
Simplify ——
4
Equation at the end of step 2 :
t3 6t
(((5 • ——) • t) - 48t) - ——
4 5
Step 3 :
Rewriting the whole as an Equivalent Fraction :
3.1 Subtracting a whole from a fraction
Rewrite the whole as a fraction using 4 as the denominator :
48t 48t • 4
48t = ——— = ———————
1 4
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 :
3.2 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:
5t4 - (48t • 4) 5t4 - 192t
——————————————— = ——————————
4 4
Equation at the end of step 3 :
(5t4 - 192t) 6t
———————————— - ——
4 5
Step 4 :
Step 5 :
Pulling out like terms :
5.1 Pull out like factors :
5t4 - 192t = t • (5t3 - 192)
Trying to factor as a Difference of Cubes:
5.2 Factoring: 5t3 - 192
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 : 5 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(t) = 5t3 - 192
Polynomial Roots Calculator is a set of methods aimed at finding values of t for which F(t)=0
Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers t 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 5 and the Trailing Constant is -192.
The factor(s) are:
of the Leading Coefficient : 1,5
of the Trailing Constant : 1 ,2 ,3 ,4 ,6 ,8 ,12 ,16 ,24 ,32 , etc
Let us test ....
P Q P/Q F(P/Q) Divisor
-1 1 -1.00 -197.00
-1 5 -0.20 -192.04
-2 1 -2.00 -232.00
-2 5 -0.40 -192.32
-3 1 -3.00 -327.00
Note - For tidiness, printing of 35 checks which found no root was suppressed
Polynomial Roots Calculator found no rational roots
Calculating the Least Common Multiple :
5.4 Find the Least Common Multiple
The left denominator is : 4
The right denominator is : 5
Number of times each prime factor
appears in the factorization of:
Prime
Factor Left
Denominator Right
Denominator L.C.M = Max
{Left,Right}
2 2 0 2
5 0 1 1
Product of all
Prime Factors 4 5 20
Least Common Multiple:
20
Calculating Multipliers :
5.5 Calculate multipliers for the two fractions
Denote the Least Common Multiple by L.C.M
Denote the Left Multiplier by Left_M
Denote the Right Multiplier by Right_M
Denote the Left Deniminator by L_Deno
Denote the Right Multiplier by R_Deno
Left_M = L.C.M / L_Deno = 5
Right_M = L.C.M / R_Deno = 4
Making Equivalent Fractions :
5.6 Rewrite the two fractions into equivalent fractions
Two fractions are called equivalent if they have the same numeric value.
For example : 1/2 and 2/4 are equivalent, y/(y+1)2 and (y2+y)/(y+1)3 are equivalent as well.
To calculate equivalent fraction , multiply the Numerator of each fraction, by its respective Multiplier.
L. Mult. • L. Num. t • (5t3-192) • 5
—————————————————— = —————————————————
L.C.M 20
R. Mult. • R. Num. 6t • 4
—————————————————— = ——————
L.C.M 20
Adding fractions that have a common denominator :
5.7 Adding up the two equivalent fractions
t • (5t3-192) • 5 - (6t • 4) 25t4 - 984t
———————————————————————————— = ———————————
20 20
Step 6 :
Pulling out like terms :
6.1 Pull out like factors :
25t4 - 984t = t • (25t3 - 984)
Trying to factor as a Difference of Cubes:
6.2 Factoring: 25t3 - 984
Check : 25 is not a cube !!
Ruling : Binomial can not be factored as the difference of two perfect cubes
Polynomial Roots Calculator :
6.3 Find roots (zeroes) of : F(t) = 25t3 - 984
See theory in step 5.3
In this case, the Leading Coefficient is 25 and the Trailing Constant is -984.
The factor(s) are:
of the Leading Coefficient : 1,5 ,25
of the Trailing Constant : 1 ,2 ,3 ,4 ,6 ,8 ,12 ,24 ,41 ,82 , etc
Let us test ....
P Q P/Q F(P/Q) Divisor
-1 1 -1.00 -1009.00
-1 5 -0.20 -984.20
-1 25 -0.04 -984.00
-2 1 -2.00 -1184.00
-2 5 -0.40 -985.60
Note - For tidiness, printing of 55 checks which found no root was suppressed
Polynomial Roots Calculator found no rational roots
Final result :
t • (25t3 - 984)
————————————————
20
Processing ends successfully