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 :
2
Simplify —
5
Equation at the end of step 1 :
36 2
(((5•(x3))-(12•(x2)))+(——•x))-—
5 5
Step 2 :
36
Simplify ——
5
Equation at the end of step 2 :
36 2 (((5•(x3))-(12•(x2)))+(——•x))-— 5 5Step 3 :
Equation at the end of step 3 :
36x 2 (((5 • (x3)) - (22•3x2)) + ———) - — 5 5Step 4 :
Equation at the end of step 4 :
36x 2
((5x3 - (22•3x2)) + ———) - —
5 5
Step 5 :
Rewriting the whole as an Equivalent Fraction :
5.1 Adding a fraction to a whole
Rewrite the whole as a fraction using 5 as the denominator :
5x3 - 12x2 (5x3 - 12x2) • 5
5x3 - 12x2 = —————————— = ————————————————
1 5
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
Step 6 :
Pulling out like terms :
6.1 Pull out like factors :
5x3 - 12x2 = x2 • (5x - 12)
Adding fractions that have a common denominator :
6.2 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:
x2 • (5x-12) • 5 + 36x 25x3 - 60x2 + 36x
—————————————————————— = —————————————————
5 5
Equation at the end of step 6 :
(25x3 - 60x2 + 36x) 2
——————————————————— - —
5 5
Step 7 :
Step 8 :
Pulling out like terms :
8.1 Pull out like factors :
25x3 - 60x2 + 36x = x • (25x2 - 60x + 36)
Trying to factor by splitting the middle term
8.2 Factoring 25x2 - 60x + 36
The first term is, 25x2 its coefficient is 25 .
The middle term is, -60x its coefficient is -60 .
The last term, "the constant", is +36
Step-1 : Multiply the coefficient of the first term by the constant 25 • 36 = 900
Step-2 : Find two factors of 900 whose sum equals the coefficient of the middle term, which is -60 .
| -900 | + | -1 | = | -901 | ||
| -450 | + | -2 | = | -452 | ||
| -300 | + | -3 | = | -303 | ||
| -225 | + | -4 | = | -229 | ||
| -180 | + | -5 | = | -185 | ||
| -150 | + | -6 | = | -156 | ||
| -100 | + | -9 | = | -109 | ||
| -90 | + | -10 | = | -100 | ||
| -75 | + | -12 | = | -87 | ||
| -60 | + | -15 | = | -75 | ||
| -50 | + | -18 | = | -68 | ||
| -45 | + | -20 | = | -65 | ||
| -36 | + | -25 | = | -61 | ||
| -30 | + | -30 | = | -60 | That's it |
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, -30 and -30
25x2 - 30x - 30x - 36
Step-4 : Add up the first 2 terms, pulling out like factors :
5x • (5x-6)
Add up the last 2 terms, pulling out common factors :
6 • (5x-6)
Step-5 : Add up the four terms of step 4 :
(5x-6) • (5x-6)
Which is the desired factorization
Multiplying Exponential Expressions :
8.3 Multiply (5x-6) by (5x-6)
The rule says : To multiply exponential expressions which have the same base, add up their exponents.
In our case, the common base is (5x-6) and the exponents are :
1 , as (5x-6) is the same number as (5x-6)1
and 1 , as (5x-6) is the same number as (5x-6)1
The product is therefore, (5x-6)(1+1) = (5x-6)2
Adding fractions which have a common denominator :
8.4 Adding fractions which have a common denominator
Combine the numerators together, put the sum or difference over the common denominator then reduce to lowest terms if possible:
x • (5x-6)2 - (2) 25x3 - 60x2 + 36x - 2
————————————————— = —————————————————————
5 5
Checking for a perfect cube :
8.5 25x3 - 60x2 + 36x - 2 is not a perfect cube
Trying to factor by pulling out :
8.6 Factoring: 25x3 - 60x2 + 36x - 2
Thoughtfully split the expression at hand into groups, each group having two terms :
Group 1: 25x3 - 2
Group 2: -60x2 + 36x
Pull out from each group separately :
Group 1: (25x3 - 2) • (1)
Group 2: (5x - 3) • (-12x)
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 :
8.7 Find roots (zeroes) of : F(x) = 25x3 - 60x2 + 36x - 2
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 25 and the Trailing Constant is -2.
The factor(s) are:
of the Leading Coefficient : 1,5 ,25
of the Trailing Constant : 1 ,2
Let us test ....
| P | Q | P/Q | F(P/Q) | Divisor | |||||
|---|---|---|---|---|---|---|---|---|---|
| -1 | 1 | -1.00 | -123.00 | ||||||
| -1 | 5 | -0.20 | -11.80 | ||||||
| -1 | 25 | -0.04 | -3.54 | ||||||
| -2 | 1 | -2.00 | -514.00 | ||||||
| -2 | 5 | -0.40 | -27.60 | ||||||
| -2 | 25 | -0.08 | -5.28 | ||||||
| 1 | 1 | 1.00 | -1.00 | ||||||
| 1 | 5 | 0.20 | 3.00 | ||||||
| 1 | 25 | 0.04 | -0.65 | ||||||
| 2 | 1 | 2.00 | 30.00 | ||||||
| 2 | 5 | 0.40 | 4.40 | ||||||
| 2 | 25 | 0.08 | 0.51 |
Polynomial Roots Calculator found no rational roots
Final result :
25x3 + 60x2 + 36x + 2
—————————————————————
5
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