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 :
(((3 • (x3)) - 2x2) - 19x) - 6Step 2 :
Equation at the end of step 2 :
((3x3 - 2x2) - 19x) - 6
Step 3 :
Checking for a perfect cube :
3.1 3x3-2x2-19x-6 is not a perfect cube
Trying to factor by pulling out :
3.2 Factoring: 3x3-2x2-19x-6
Thoughtfully split the expression at hand into groups, each group having two terms :
Group 1: -19x-6
Group 2: 3x3-2x2
Pull out from each group separately :
Group 1: (19x+6) • (-1)
Group 2: (3x-2) • (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 :
3.3 Find roots (zeroes) of : F(x) = 3x3-2x2-19x-6
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 3 and the Trailing Constant is -6.
The factor(s) are:
of the Leading Coefficient : 1,3
of the Trailing Constant : 1 ,2 ,3 ,6
Let us test ....
| P | Q | P/Q | F(P/Q) | Divisor | |||||
|---|---|---|---|---|---|---|---|---|---|
| -1 | 1 | -1.00 | 8.00 | ||||||
| -1 | 3 | -0.33 | 0.00 | 3x+1 | |||||
| -2 | 1 | -2.00 | 0.00 | x+2 | |||||
| -2 | 3 | -0.67 | 4.89 | ||||||
| -3 | 1 | -3.00 | -48.00 | ||||||
| -6 | 1 | -6.00 | -612.00 | ||||||
| 1 | 1 | 1.00 | -24.00 | ||||||
| 1 | 3 | 0.33 | -12.44 | ||||||
| 2 | 1 | 2.00 | -28.00 | ||||||
| 2 | 3 | 0.67 | -18.67 | ||||||
| 3 | 1 | 3.00 | 0.00 | x-3 | |||||
| 6 | 1 | 6.00 | 456.00 |
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
3x3-2x2-19x-6
can be divided by 3 different polynomials,including by x-3
Polynomial Long Division :
3.4 Polynomial Long Division
Dividing : 3x3-2x2-19x-6
("Dividend")
By : x-3 ("Divisor")
| dividend | 3x3 | - | 2x2 | - | 19x | - | 6 | ||
| - divisor | * 3x2 | 3x3 | - | 9x2 | |||||
| remainder | 7x2 | - | 19x | - | 6 | ||||
| - divisor | * 7x1 | 7x2 | - | 21x | |||||
| remainder | 2x | - | 6 | ||||||
| - divisor | * 2x0 | 2x | - | 6 | |||||
| remainder | 0 |
Quotient : 3x2+7x+2 Remainder: 0
Trying to factor by splitting the middle term
3.5 Factoring 3x2+7x+2
The first term is, 3x2 its coefficient is 3 .
The middle term is, +7x its coefficient is 7 .
The last term, "the constant", is +2
Step-1 : Multiply the coefficient of the first term by the constant 3 • 2 = 6
Step-2 : Find two factors of 6 whose sum equals the coefficient of the middle term, which is 7 .
| -6 | + | -1 | = | -7 | ||
| -3 | + | -2 | = | -5 | ||
| -2 | + | -3 | = | -5 | ||
| -1 | + | -6 | = | -7 | ||
| 1 | + | 6 | = | 7 | That's it |
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, 1 and 6
3x2 + 1x + 6x + 2
Step-4 : Add up the first 2 terms, pulling out like factors :
x • (3x+1)
Add up the last 2 terms, pulling out common factors :
2 • (3x+1)
Step-5 : Add up the four terms of step 4 :
(x+2) • (3x+1)
Which is the desired factorization
Final result :
(3x + 1) • (x + 2) • (x - 3)
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