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•3x2)) - 63x) + 108
Step 2 :
Checking for a perfect cube :
2.1 x3+6x2-63x+108 is not a perfect cube
Trying to factor by pulling out :
2.2 Factoring: x3+6x2-63x+108
Thoughtfully split the expression at hand into groups, each group having two terms :
Group 1: -63x+108
Group 2: 6x2+x3
Pull out from each group separately :
Group 1: (7x-12) • (-9)
Group 2: (x+6) • (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 :
2.3 Find roots (zeroes) of : F(x) = x3+6x2-63x+108
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 108.
The factor(s) are:
of the Leading Coefficient : 1
of the Trailing Constant : 1 ,2 ,3 ,4 ,6 ,9 ,12 ,18 ,27 ,36 , etc
Let us test ....
| P | Q | P/Q | F(P/Q) | Divisor | |||||
|---|---|---|---|---|---|---|---|---|---|
| -1 | 1 | -1.00 | 176.00 | ||||||
| -2 | 1 | -2.00 | 250.00 | ||||||
| -3 | 1 | -3.00 | 324.00 | ||||||
| -4 | 1 | -4.00 | 392.00 | ||||||
| -6 | 1 | -6.00 | 486.00 | ||||||
| -9 | 1 | -9.00 | 432.00 | ||||||
| -12 | 1 | -12.00 | 0.00 | x+12 | |||||
| -18 | 1 | -18.00 | -2646.00 | ||||||
| -27 | 1 | -27.00 | -13500.00 | ||||||
| -36 | 1 | -36.00 | -36504.00 | ||||||
| 1 | 1 | 1.00 | 52.00 | ||||||
| 2 | 1 | 2.00 | 14.00 | ||||||
| 3 | 1 | 3.00 | 0.00 | x-3 | |||||
| 4 | 1 | 4.00 | 16.00 | ||||||
| 6 | 1 | 6.00 | 162.00 | ||||||
| 9 | 1 | 9.00 | 756.00 | ||||||
| 12 | 1 | 12.00 | 1944.00 | ||||||
| 18 | 1 | 18.00 | 6750.00 | ||||||
| 27 | 1 | 27.00 | 22464.00 | ||||||
| 36 | 1 | 36.00 | 52272.00 |
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+6x2-63x+108
can be divided by 2 different polynomials,including by x-3
Polynomial Long Division :
2.4 Polynomial Long Division
Dividing : x3+6x2-63x+108
("Dividend")
By : x-3 ("Divisor")
| dividend | x3 | + | 6x2 | - | 63x | + | 108 | ||
| - divisor | * x2 | x3 | - | 3x2 | |||||
| remainder | 9x2 | - | 63x | + | 108 | ||||
| - divisor | * 9x1 | 9x2 | - | 27x | |||||
| remainder | - | 36x | + | 108 | |||||
| - divisor | * -36x0 | - | 36x | + | 108 | ||||
| remainder | 0 |
Quotient : x2+9x-36 Remainder: 0
Trying to factor by splitting the middle term
2.5 Factoring x2+9x-36
The first term is, x2 its coefficient is 1 .
The middle term is, +9x its coefficient is 9 .
The last term, "the constant", is -36
Step-1 : Multiply the coefficient of the first term by the constant 1 • -36 = -36
Step-2 : Find two factors of -36 whose sum equals the coefficient of the middle term, which is 9 .
| -36 | + | 1 | = | -35 | ||
| -18 | + | 2 | = | -16 | ||
| -12 | + | 3 | = | -9 | ||
| -9 | + | 4 | = | -5 | ||
| -6 | + | 6 | = | 0 | ||
| -4 | + | 9 | = | 5 | ||
| -3 | + | 12 | = | 9 | That's it |
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, -3 and 12
x2 - 3x + 12x - 36
Step-4 : Add up the first 2 terms, pulling out like factors :
x • (x-3)
Add up the last 2 terms, pulling out common factors :
12 • (x-3)
Step-5 : Add up the four terms of step 4 :
(x+12) • (x-3)
Which is the desired factorization
Multiplying Exponential Expressions :
2.6 Multiply (x-3) by (x-3)
The rule says : To multiply exponential expressions which have the same base, add up their exponents.
In our case, the common base is (x-3) and the exponents are :
1 , as (x-3) is the same number as (x-3)1
and 1 , as (x-3) is the same number as (x-3)1
The product is therefore, (x-3)(1+1) = (x-3)2
Final result :
(x + 12) • (x - 3)2
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