Solution - Adding, subtracting and finding the least common multiple
Other Ways to Solve
Adding, subtracting and finding the least common multipleStep by Step Solution
Step 1 :
1
Simplify ——
x2
Equation at the end of step 1 :
x 1 ((((————-16)-x)+——)-5x)+4 (x2) x2Step 2 :
x Simplify —— x2
Dividing exponential expressions :
2.1 x1 divided by x2 = x(1 - 2) = x(-1) = 1/x1 = 1/x
Equation at the end of step 2 :
1 1
((((— - 16) - x) + ——) - 5x) + 4
x 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 x as the denominator :
16 16 • x
16 = —— = ——————
1 x
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:
1 - (16 • x) 1 - 16x
———————————— = ———————
x x
Equation at the end of step 3 :
(1 - 16x) 1
(((————————— - x) + ——) - 5x) + 4
x x2
Step 4 :
Rewriting the whole as an Equivalent Fraction :
4.1 Subtracting a whole from a fraction
Rewrite the whole as a fraction using x as the denominator :
x x • x
x = — = —————
1 x
Adding fractions that have a common denominator :
4.2 Adding up the two equivalent fractions
(1-16x) - (x • x) -x2 - 16x + 1
————————————————— = —————————————
x x
Equation at the end of step 4 :
(-x2 - 16x + 1) 1
((——————————————— + ——) - 5x) + 4
x x2
Step 5 :
Step 6 :
Pulling out like terms :
6.1 Pull out like factors :
-x2 - 16x + 1 = -1 • (x2 + 16x - 1)
Trying to factor by splitting the middle term
6.2 Factoring x2 + 16x - 1
The first term is, x2 its coefficient is 1 .
The middle term is, +16x its coefficient is 16 .
The last term, "the constant", is -1
Step-1 : Multiply the coefficient of the first term by the constant 1 • -1 = -1
Step-2 : Find two factors of -1 whose sum equals the coefficient of the middle term, which is 16 .
| -1 | + | 1 | = | 0 |
Observation : No two such factors can be found !!
Conclusion : Trinomial can not be factored
Calculating the Least Common Multiple :
6.3 Find the Least Common Multiple
The left denominator is : x
The right denominator is : x2
| Algebraic Factor | Left Denominator | Right Denominator | L.C.M = Max {Left,Right} |
|---|---|---|---|
| x | 1 | 2 | 2 |
Least Common Multiple:
x2
Calculating Multipliers :
6.4 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 = x
Right_M = L.C.M / R_Deno = 1
Making Equivalent Fractions :
6.5 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.
To calculate equivalent fraction , multiply the Numerator of each fraction, by its respective Multiplier.
L. Mult. • L. Num. (-x2-16x+1) • x —————————————————— = ——————————————— L.C.M x2 R. Mult. • R. Num. 1 —————————————————— = —— L.C.M x2
Adding fractions that have a common denominator :
6.6 Adding up the two equivalent fractions
(-x2-16x+1) • x + 1 -x3 - 16x2 + x + 1
——————————————————— = ——————————————————
x2 x2
Equation at the end of step 6 :
(-x3 - 16x2 + x + 1)
(———————————————————— - 5x) + 4
x2
Step 7 :
Rewriting the whole as an Equivalent Fraction :
7.1 Subtracting a whole from a fraction
Rewrite the whole as a fraction using x2 as the denominator :
5x 5x • x2
5x = —— = ———————
1 x2
Checking for a perfect cube :
7.2 -x3 - 16x2 + x + 1 is not a perfect cube
Trying to factor by pulling out :
7.3 Factoring: -x3 - 16x2 + x + 1
Thoughtfully split the expression at hand into groups, each group having two terms :
Group 1: x + 1
Group 2: -x3 - 16x2
Pull out from each group separately :
Group 1: (x + 1) • (1)
Group 2: (x + 16) • (-x2)
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 :
7.4 Find roots (zeroes) of : F(x) = -x3 - 16x2 + x + 1
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 1.
The factor(s) are:
of the Leading Coefficient : 1
of the Trailing Constant : 1
Let us test ....
| P | Q | P/Q | F(P/Q) | Divisor | |||||
|---|---|---|---|---|---|---|---|---|---|
| -1 | 1 | -1.00 | -15.00 | ||||||
| 1 | 1 | 1.00 | -15.00 |
Polynomial Roots Calculator found no rational roots
Adding fractions that have a common denominator :
7.5 Adding up the two equivalent fractions
(-x3-16x2+x+1) - (5x • x2) -6x3 - 16x2 + x + 1
—————————————————————————— = ———————————————————
x2 x2
Equation at the end of step 7 :
(-6x3 - 16x2 + x + 1)
————————————————————— + 4
x2
Step 8 :
Rewriting the whole as an Equivalent Fraction :
8.1 Adding a whole to a fraction
Rewrite the whole as a fraction using x2 as the denominator :
4 4 • x2
4 = — = ——————
1 x2
Checking for a perfect cube :
8.2 -6x3 - 16x2 + x + 1 is not a perfect cube
Trying to factor by pulling out :
8.3 Factoring: -6x3 - 16x2 + x + 1
Thoughtfully split the expression at hand into groups, each group having two terms :
Group 1: x + 1
Group 2: -6x3 - 16x2
Pull out from each group separately :
Group 1: (x + 1) • (1)
Group 2: (3x + 8) • (-2x2)
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 :
8.4 Find roots (zeroes) of : F(x) = -6x3 - 16x2 + x + 1
See theory in step 7.4
In this case, the Leading Coefficient is -6 and the Trailing Constant is 1.
The factor(s) are:
of the Leading Coefficient : 1,2 ,3 ,6
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 | -2.75 | ||||||
| -1 | 3 | -0.33 | -0.89 | ||||||
| -1 | 6 | -0.17 | 0.42 | ||||||
| 1 | 1 | 1.00 | -20.00 | ||||||
| 1 | 2 | 0.50 | -3.25 | ||||||
| 1 | 3 | 0.33 | -0.67 | ||||||
| 1 | 6 | 0.17 | 0.69 |
Polynomial Roots Calculator found no rational roots
Adding fractions that have a common denominator :
8.5 Adding up the two equivalent fractions
(-6x3-16x2+x+1) + 4 • x2 -6x3 - 12x2 + x + 1
———————————————————————— = ———————————————————
x2 x2
Checking for a perfect cube :
8.6 -6x3 - 12x2 + x + 1 is not a perfect cube
Trying to factor by pulling out :
8.7 Factoring: -6x3 - 12x2 + x + 1
Thoughtfully split the expression at hand into groups, each group having two terms :
Group 1: x + 1
Group 2: -6x3 - 12x2
Pull out from each group separately :
Group 1: (x + 1) • (1)
Group 2: (x + 2) • (-6x2)
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 :
8.8 Find roots (zeroes) of : F(x) = -6x3 - 12x2 + x + 1
See theory in step 7.4
In this case, the Leading Coefficient is -6 and the Trailing Constant is 1.
The factor(s) are:
of the Leading Coefficient : 1,2 ,3 ,6
of the Trailing Constant : 1
Let us test ....
| P | Q | P/Q | F(P/Q) | Divisor | |||||
|---|---|---|---|---|---|---|---|---|---|
| -1 | 1 | -1.00 | -6.00 | ||||||
| -1 | 2 | -0.50 | -1.75 | ||||||
| -1 | 3 | -0.33 | -0.44 | ||||||
| -1 | 6 | -0.17 | 0.53 | ||||||
| 1 | 1 | 1.00 | -16.00 | ||||||
| 1 | 2 | 0.50 | -2.25 | ||||||
| 1 | 3 | 0.33 | -0.22 | ||||||
| 1 | 6 | 0.17 | 0.81 |
Polynomial Roots Calculator found no rational roots
Final result :
6x3 - 12x2 + x + 1
———————————————————
x2
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