Factorise 16a³ - 2/a³ + 6a - 3/a
Answers
Step-by-step explanation:
STEP
1
:
3
Simplify —
a
Equation at the end of step
1
:
2 3
(((16•(a3))-————)+6a)-—
(a3) a
STEP
2
:
2
Simplify ——
a3
Equation at the end of step
2
:
2 3
(((16 • (a3)) - ——) + 6a) - —
a3 a
STEP
3
:
Equation at the end of step
3
:
2 3
((24a3 - ——) + 6a) - —
a3 a
STEP
4
:
Rewriting the whole as an Equivalent Fraction
4.1 Subtracting a fraction from a whole
Rewrite the whole as a fraction using a3 as the denominator :
24a3 24a3 • a3
24a3 = ———— = —————————
1 a3
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 :
4.2 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:
24a3 • a3 - (2) 16a6 - 2
——————————————— = ————————
a3 a3
Equation at the end of step
4
:
(16a6 - 2) 3
(—————————— + 6a) - —
a3 a
STEP
5
:
Rewriting the whole as an Equivalent Fraction :
5.1 Adding a whole to a fraction
Rewrite the whole as a fraction using a3 as the denominator :
6a 6a • a3
6a = —— = ———————
1 a3
STEP
6
:
Pulling out like terms :
6.1 Pull out like factors :
16a6 - 2 = 2 • (8a6 - 1)
Trying to factor as a Difference of Squares:
6.2 Factoring: 8a6 - 1
Theory : A difference of two perfect squares, A2 - B2 can be factored into (A+B) • (A-B)
Proof : (A+B) • (A-B) =
A2 - AB + BA - B2 =
A2 - AB + AB - B2 =
A2 - B2
Note : AB = BA is the commutative property of multiplication.
Note : - AB + AB equals zero and is therefore eliminated from the expression.
Check : 8 is not a square !!
Ruling : Binomial can not be factored as the
difference of two perfect squares
Polynomial Roots Calculator :
6.3 Find roots (zeroes) of : F(a) = 8a6 - 1
Polynomial Roots Calculator is a set of methods aimed at finding values of a for which F(a)=0
Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers a 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 8 and the Trailing Constant is -1.
The factor(s) are:
of the Leading Coefficient : 1,2 ,4 ,8
of the Trailing Constant : 1
Let us test ....
P Q P/Q F(P/Q) Divisor
-1 1 -1.00 7.00
-1 2 -0.50 -0.88
-1 4 -0.25 -1.00
-1 8 -0.12 -1.00
1 1 1.00 7.00
1 2 0.50 -0.88
1 4 0.25 -1.00
1 8 0.12 -1.00
Polynomial Roots Calculator found no rational roots
Adding fractions that have a common denominator :
6.4 Adding up the two equivalent fractions
2 • (8a6-1) + 6a • a3 16a6 + 6a4 - 2
————————————————————— = ——————————————
a3 a3
Equation at the end of step
6
:
(16a6 + 6a4 - 2) 3
———————————————— - —
a3 a
STEP
7
:
STEP
8
:
Pulling out like terms :
8.1 Pull out like factors :
16a6 + 6a4 - 2 = 2 • (8a6 + 3a4 - 1)
Polynomial Roots Calculator :
8.2 Find roots (zeroes) of : F(a) = 8a6 + 3a4 - 1
See theory in step 6.3
In this case, the Leading Coefficient is 8 and the Trailing Constant is -1.
The factor(s) are:
of the Leading Coefficient : 1,2 ,4 ,8
of the Trailing Constant : 1
Let us test ....
P Q P/Q F(P/Q) Divisor
-1 1 -1.00 10.00
-1 2 -0.50 -0.69
-1 4 -0.25 -0.99
-1 8 -0.12 -1.00
1 1 1.00 10.00
1 2 0.50 -0.69
1 4 0.25 -0.99
1 8 0.12 -1.00
Polynomial Roots Calculator found no rational roots
Calculating the Least Common Multiple :
8.3 Find the Least Common Multiple
The left denominator is : a3
The right denominator is : a
Number of times each Algebraic Factor
appears in the factorization of:
Algebraic
Factor Left
Denominator Right
Denominator L.C.M = Max
{Left,Right}
a 3 1 3
Least Common Multiple:
a3
Calculating Multipliers :
8.4 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 = 1
Right_M = L.C.M / R_Deno = a2
Making Equivalent Fractions :
8.5 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.
L. Mult. • L. Num. 2 • (8a6+3a4-1)
—————————————————— = ———————————————
L.C.M a3
R. Mult. • R. Num. 3 • a2
—————————————————— = ——————
L.C.M a3
2 • (8a6+3a4-1) - (3 • a2) 16a6 + 6a4 - 3a2 - 2
—————————————————————————— = ————————————————————
a3 a3
Final result :
16a6 + 6a4 3a2 - 2
————————————————————
a3
Answer:
16a³- 1 . 2/a³+6a - 1 . 3/a