Solution - Reducing fractions to their lowest terms
Other Ways to Solve
Reducing fractions to their lowest termsStep by Step Solution
Step 1 :
9
Simplify ——
a2
Equation at the end of step 1 :
3 9 (((a+————)+a)-12)——)+7a)+12) (a2)((((a^2)a2Step 2 :
Rewriting the whole as an Equivalent Fraction :
2.1 Subtracting a fraction from a whole
Rewrite the whole as a fraction using a2 as the denominator :
a2 a2 • a2
a2 = —— = ———————
1 a2
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:
a2 • a2 - (9) a4 - 9
————————————— = ——————
a2 a2
Equation at the end of step 2 :
3 (a4-9)
(((a+————)+a)-12)——————+7a)+12)
(a2)(( a2
Step 3 :
Rewriting the whole as an Equivalent Fraction :
3.1 Adding a whole to a fraction
Rewrite the whole as a fraction using a2 as the denominator :
7a 7a • a2
7a = —— = ———————
1 a2
Trying to factor as a Difference of Squares :
3.2 Factoring: a4 - 9
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 : 9 is the square of 3
Check : a4 is the square of a2
Factorization is : (a2 + 3) • (a2 - 3)
Polynomial Roots Calculator :
3.3 Find roots (zeroes) of : F(a) = a2 + 3
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 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 | 12.00 | ||||||
| 1 | 1 | 1.00 | 4.00 | ||||||
| 3 | 1 | 3.00 | 12.00 |
Polynomial Roots Calculator found no rational roots
Trying to factor as a Difference of Squares :
3.4 Factoring: a2 - 3
Check : 3 is not a square !!
Ruling : Binomial can not be factored as the difference of two perfect squares.
Adding fractions that have a common denominator :
3.5 Adding up the two equivalent fractions
(a2+3) • (a2-3) + 7a • a2 a4 + 7a3 - 9
————————————————————————— = ————————————
a2 a2
Equation at the end of step 3 :
3 (a4+7a3-9)
(((a+————)+a)-12)——————————+12)
(a2)( a2
Step 4 :
Rewriting the whole as an Equivalent Fraction :
4.1 Adding a whole to a fraction
Rewrite the whole as a fraction using a2 as the denominator :
12 12 • a2
12 = —— = ———————
1 a2
Polynomial Roots Calculator :
4.2 Find roots (zeroes) of : F(a) = a4 + 7a3 - 9
See theory in step 3.3
In this case, the Leading Coefficient is 1 and the Trailing Constant is -9.
The factor(s) are:
of the Leading Coefficient : 1
of the Trailing Constant : 1 ,3 ,9
Let us test ....
| P | Q | P/Q | F(P/Q) | Divisor | |||||
|---|---|---|---|---|---|---|---|---|---|
| -1 | 1 | -1.00 | -15.00 | ||||||
| -3 | 1 | -3.00 | -117.00 | ||||||
| -9 | 1 | -9.00 | 1449.00 | ||||||
| 1 | 1 | 1.00 | -1.00 | ||||||
| 3 | 1 | 3.00 | 261.00 | ||||||
| 9 | 1 | 9.00 | 11655.00 |
Polynomial Roots Calculator found no rational roots
Adding fractions that have a common denominator :
4.3 Adding up the two equivalent fractions
(a4+7a3-9) + 12 • a2 a4 + 7a3 + 12a2 - 9
———————————————————— = ———————————————————
a2 a2
Equation at the end of step 4 :
3 (a4+7a3+12a2-9) (((a+————)+a)-12)——————————————— (a2) a2Step 5 :
3 Simplify —— a2
Equation at the end of step 5 :
3 (a4 + 7a3 + 12a2 - 9)
——) + a) - 12) ÷ —————————————————————
a2 a2
Step 6 :
Rewriting the whole as an Equivalent Fraction :
6.1 Adding a fraction to a whole
Rewrite the whole as a fraction using a2 as the denominator :
a a • a2
a = — = ——————
1 a2
Adding fractions that have a common denominator :
6.2 Adding up the two equivalent fractions
a • a2 + 3 a3 + 3
—————————— = ——————
a2 a2
Equation at the end of step 6 :
(a3 + 3) (a4 + 7a3 + 12a2 - 9)
———————— + a) - 12) ÷ —————————————————————
a2 a2
Step 7 :
Rewriting the whole as an Equivalent Fraction :
7.1 Adding a whole to a fraction
Rewrite the whole as a fraction using a2 as the denominator :
a a • a2
a = — = ——————
1 a2
Trying to factor as a Sum of Cubes :
7.2 Factoring: a3 + 3
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 : 3 is not a cube !!
Ruling : Binomial can not be factored as the difference of two perfect cubes
Polynomial Roots Calculator :
7.3 Find roots (zeroes) of : F(a) = a3 + 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 | 2.00 | ||||||
| -3 | 1 | -3.00 | -24.00 | ||||||
| 1 | 1 | 1.00 | 4.00 | ||||||
| 3 | 1 | 3.00 | 30.00 |
Polynomial Roots Calculator found no rational roots
Adding fractions that have a common denominator :
7.4 Adding up the two equivalent fractions
(a3+3) + a • a2 2a3 + 3
——————————————— = ———————
a2 a2
Equation at the end of step 7 :
(2a3 + 3) (a4 + 7a3 + 12a2 - 9)
————————— - 12) ÷ —————————————————————
a2 a2
Step 8 :
Rewriting the whole as an Equivalent Fraction :
8.1 Subtracting a whole from a fraction
Rewrite the whole as a fraction using a2 as the denominator :
12 12 • a2
12 = —— = ———————
1 a2
Trying to factor as a Sum of Cubes :
8.2 Factoring: 2a3 + 3
Check : 2 is not a cube !!
Ruling : Binomial can not be factored as the difference of two perfect cubes
Polynomial Roots Calculator :
8.3 Find roots (zeroes) of : F(a) = 2a3 + 3
See theory in step 3.3
In this case, the Leading Coefficient is 2 and the Trailing Constant is 3.
The factor(s) are:
of the Leading Coefficient : 1,2
of the Trailing Constant : 1 ,3
Let us test ....
| P | Q | P/Q | F(P/Q) | Divisor | |||||
|---|---|---|---|---|---|---|---|---|---|
| -1 | 1 | -1.00 | 1.00 | ||||||
| -1 | 2 | -0.50 | 2.75 | ||||||
| -3 | 1 | -3.00 | -51.00 | ||||||
| -3 | 2 | -1.50 | -3.75 | ||||||
| 1 | 1 | 1.00 | 5.00 | ||||||
| 1 | 2 | 0.50 | 3.25 | ||||||
| 3 | 1 | 3.00 | 57.00 | ||||||
| 3 | 2 | 1.50 | 9.75 |
Polynomial Roots Calculator found no rational roots
Adding fractions that have a common denominator :
8.4 Adding up the two equivalent fractions
(2a3+3) - (12 • a2) 2a3 - 12a2 + 3
——————————————————— = ——————————————
a2 a2
Equation at the end of step 8 :
(2a3 - 12a2 + 3) (a4 + 7a3 + 12a2 - 9)
———————————————— ÷ —————————————————————
a2 a2
Step 9 :
2a3-12a2+3 a4+7a3+12a2-9
Divide —————————— by —————————————
a2 a2
9.1 Dividing fractions
To divide fractions, write the divison as multiplication by the reciprocal of the divisor :
2a3 - 12a2 + 3 a4 + 7a3 + 12a2 - 9 2a3 - 12a2 + 3 a2 —————————————— ÷ ——————————————————— = —————————————— • ————————————————————— a2 a2 a2 (a4 + 7a3 + 12a2 - 9)
Checking for a perfect cube :
9.2 a4 + 7a3 + 12a2 - 9 is not a perfect cube
Trying to factor by pulling out :
9.3 Factoring: a4 + 7a3 + 12a2 - 9
Thoughtfully split the expression at hand into groups, each group having two terms :
Group 1: 12a2 - 9
Group 2: a4 + 7a3
Pull out from each group separately :
Group 1: (4a2 - 3) • (3)
Group 2: (a + 7) • (a3)
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 :
9.4 Find roots (zeroes) of : F(a) = 2a3 - 12a2 + 3
See theory in step 3.3
In this case, the Leading Coefficient is 2 and the Trailing Constant is 3.
The factor(s) are:
of the Leading Coefficient : 1,2
of the Trailing Constant : 1 ,3
Let us test ....
| P | Q | P/Q | F(P/Q) | Divisor | |||||
|---|---|---|---|---|---|---|---|---|---|
| -1 | 1 | -1.00 | -11.00 | ||||||
| -1 | 2 | -0.50 | -0.25 | ||||||
| -3 | 1 | -3.00 | -159.00 | ||||||
| -3 | 2 | -1.50 | -30.75 | ||||||
| 1 | 1 | 1.00 | -7.00 | ||||||
| 1 | 2 | 0.50 | 0.25 | ||||||
| 3 | 1 | 3.00 | -51.00 | ||||||
| 3 | 2 | 1.50 | -17.25 |
Polynomial Roots Calculator found no rational roots
Polynomial Roots Calculator :
9.5 Find roots (zeroes) of : F(a) = a4 + 7a3 + 12a2 - 9
See theory in step 3.3
In this case, the Leading Coefficient is 1 and the Trailing Constant is -9.
The factor(s) are:
of the Leading Coefficient : 1
of the Trailing Constant : 1 ,3 ,9
Let us test ....
| P | Q | P/Q | F(P/Q) | Divisor | |||||
|---|---|---|---|---|---|---|---|---|---|
| -1 | 1 | -1.00 | -3.00 | ||||||
| -3 | 1 | -3.00 | -9.00 | ||||||
| -9 | 1 | -9.00 | 2421.00 | ||||||
| 1 | 1 | 1.00 | 11.00 | ||||||
| 3 | 1 | 3.00 | 369.00 | ||||||
| 9 | 1 | 9.00 | 12627.00 |
Polynomial Roots Calculator found no rational roots
Final result :
2a3 - 12a2 + 3
———————————————————
a4 + 7a3 + 12a2 - 9
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