Solution - Finding the roots of polynomials
Other Ways to Solve
Finding the roots of polynomialsStep by Step Solution
Step 1 :
Equation at the end of step 1 :
(((x3) - (2•32x2)) + 99x) - 162
Step 2 :
Checking for a perfect cube :
2.1 x3-18x2+99x-162 is not a perfect cube
Trying to factor by pulling out :
2.2 Factoring: x3-18x2+99x-162
Thoughtfully split the expression at hand into groups, each group having two terms :
Group 1: x3-162
Group 2: -18x2+99x
Pull out from each group separately :
Group 1: (x3-162) • (1)
Group 2: (2x-11) • (-9x)
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) = x3-18x2+99x-162
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 -162.
The factor(s) are:
of the Leading Coefficient : 1
of the Trailing Constant : 1 ,2 ,3 ,6 ,9 ,18 ,27 ,54 ,81 ,162
Let us test ....
| P | Q | P/Q | F(P/Q) | Divisor | |||||
|---|---|---|---|---|---|---|---|---|---|
| -1 | 1 | -1.00 | -280.00 | ||||||
| -2 | 1 | -2.00 | -440.00 | ||||||
| -3 | 1 | -3.00 | -648.00 | ||||||
| -6 | 1 | -6.00 | -1620.00 | ||||||
| -9 | 1 | -9.00 | -3240.00 | ||||||
| -18 | 1 | -18.00 | -13608.00 | ||||||
| -27 | 1 | -27.00 | -35640.00 | ||||||
| -54 | 1 | -54.00 | -215460.00 | ||||||
| -81 | 1 | -81.00 | -657720.00 | ||||||
| -162 | 1 | -162.00 | -4740120.00 | ||||||
| 1 | 1 | 1.00 | -80.00 | ||||||
| 2 | 1 | 2.00 | -28.00 | ||||||
| 3 | 1 | 3.00 | 0.00 | x-3 | |||||
| 6 | 1 | 6.00 | 0.00 | x-6 | |||||
| 9 | 1 | 9.00 | 0.00 | x-9 | |||||
| 18 | 1 | 18.00 | 1620.00 | ||||||
| 27 | 1 | 27.00 | 9072.00 | ||||||
| 54 | 1 | 54.00 | 110160.00 | ||||||
| 81 | 1 | 81.00 | 421200.00 | ||||||
| 162 | 1 | 162.00 | 3795012.00 |
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
x3-18x2+99x-162
can be divided by 3 different polynomials,including by x-9
Polynomial Long Division :
2.4 Polynomial Long Division
Dividing : x3-18x2+99x-162
("Dividend")
By : x-9 ("Divisor")
| dividend | x3 | - | 18x2 | + | 99x | - | 162 | ||
| - divisor | * x2 | x3 | - | 9x2 | |||||
| remainder | - | 9x2 | + | 99x | - | 162 | |||
| - divisor | * -9x1 | - | 9x2 | + | 81x | ||||
| remainder | 18x | - | 162 | ||||||
| - divisor | * 18x0 | 18x | - | 162 | |||||
| remainder | 0 |
Quotient : x2-9x+18 Remainder: 0
Trying to factor by splitting the middle term
2.5 Factoring x2-9x+18
The first term is, x2 its coefficient is 1 .
The middle term is, -9x its coefficient is -9 .
The last term, "the constant", is +18
Step-1 : Multiply the coefficient of the first term by the constant 1 • 18 = 18
Step-2 : Find two factors of 18 whose sum equals the coefficient of the middle term, which is -9 .
| -18 | + | -1 | = | -19 | ||
| -9 | + | -2 | = | -11 | ||
| -6 | + | -3 | = | -9 | That's it |
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, -6 and -3
x2 - 6x - 3x - 18
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 :
3 • (x-6)
Step-5 : Add up the four terms of step 4 :
(x-3) • (x-6)
Which is the desired factorization
Final result :
(x - 3) • (x - 6) • (x - 9)
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