Step by Step Solution
Reformatting the input :
Changes made to your input should not affect the solution:
(1): "x2" was replaced by "x^2". 1 more similar replacement(s).
Step 1 :
Equation at the end of step 1 :
(((x3) - (2•5x2)) + 12x) + 72
Step 2 :
Checking for a perfect cube :
2.1 x3-10x2+12x+72 is not a perfect cube
Trying to factor by pulling out :
2.2 Factoring: x3-10x2+12x+72
Thoughtfully split the expression at hand into groups, each group having two terms :
Group 1: x3-10x2
Group 2: 12x+72
Pull out from each group separately :
Group 1: (x-10) • (x2)
Group 2: (x+6) • (12)
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 :
2.3 Find roots (zeroes) of : F(x) = x3-10x2+12x+72
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 72.
The factor(s) are:
of the Leading Coefficient : 1
of the Trailing Constant : 1 ,2 ,3 ,4 ,6 ,8 ,9 ,12 ,18 ,24 , etc
Let us test ....
| P | Q | P/Q | F(P/Q) | Divisor | |||||
|---|---|---|---|---|---|---|---|---|---|
| -1 | 1 | -1.00 | 49.00 | ||||||
| -2 | 1 | -2.00 | 0.00 | x+2 | |||||
| -3 | 1 | -3.00 | -81.00 | ||||||
| -4 | 1 | -4.00 | -200.00 | ||||||
| -6 | 1 | -6.00 | -576.00 | ||||||
| -8 | 1 | -8.00 | -1176.00 | ||||||
| -9 | 1 | -9.00 | -1575.00 | ||||||
| -12 | 1 | -12.00 | -3240.00 | ||||||
| -18 | 1 | -18.00 | -9216.00 | ||||||
| -24 | 1 | -24.00 | -19800.00 | ||||||
| 1 | 1 | 1.00 | 75.00 | ||||||
| 2 | 1 | 2.00 | 64.00 | ||||||
| 3 | 1 | 3.00 | 45.00 | ||||||
| 4 | 1 | 4.00 | 24.00 | ||||||
| 6 | 1 | 6.00 | 0.00 | x-6 | |||||
| 8 | 1 | 8.00 | 40.00 | ||||||
| 9 | 1 | 9.00 | 99.00 | ||||||
| 12 | 1 | 12.00 | 504.00 | ||||||
| 18 | 1 | 18.00 | 2880.00 | ||||||
| 24 | 1 | 24.00 | 8424.00 |
The Factor Theorem states that if P/Q is root of a polynomial then this polynomial can be divided by q*x-p Note that q and p originate from P/Q reduced to its lowest terms
In our case this means that
x3-10x2+12x+72
can be divided by 2 different polynomials,including by x-6
Polynomial Long Division :
2.4 Polynomial Long Division
Dividing : x3-10x2+12x+72
("Dividend")
By : x-6 ("Divisor")
| dividend | x3 | - | 10x2 | + | 12x | + | 72 | ||
| - divisor | * x2 | x3 | - | 6x2 | |||||
| remainder | - | 4x2 | + | 12x | + | 72 | |||
| - divisor | * -4x1 | - | 4x2 | + | 24x | ||||
| remainder | - | 12x | + | 72 | |||||
| - divisor | * -12x0 | - | 12x | + | 72 | ||||
| remainder | 0 |
Quotient : x2-4x-12 Remainder: 0
Trying to factor by splitting the middle term
2.5 Factoring x2-4x-12
The first term is, x2 its coefficient is 1 .
The middle term is, -4x its coefficient is -4 .
The last term, "the constant", is -12
Step-1 : Multiply the coefficient of the first term by the constant 1 • -12 = -12
Step-2 : Find two factors of -12 whose sum equals the coefficient of the middle term, which is -4 .
| -12 | + | 1 | = | -11 | ||
| -6 | + | 2 | = | -4 | That's it |
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, -6 and 2
x2 - 6x + 2x - 12
Step-4 : Add up the first 2 terms, pulling out like factors :
x • (x-6)
Add up the last 2 terms, pulling out common factors :
2 • (x-6)
Step-5 : Add up the four terms of step 4 :
(x+2) • (x-6)
Which is the desired factorization
Multiplying Exponential Expressions :
2.6 Multiply (x-6) by (x-6)
The rule says : To multiply exponential expressions which have the same base, add up their exponents.
In our case, the common base is (x-6) and the exponents are :
1 , as (x-6) is the same number as (x-6)1
and 1 , as (x-6) is the same number as (x-6)1
The product is therefore, (x-6)(1+1) = (x-6)2
Final result :
(x + 2) • (x - 6)2
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