Solution - Finding the roots of polynomials
Other Ways to Solve
Finding the roots of polynomialsStep by Step Solution
Step 1 :
1.1 Evaluate : (3y-1)3 = 27y3-27y2+9y-1
Checking for a perfect cube :
1.2 -27y3+27y2-9y+2 is not a perfect cube
Trying to factor by pulling out :
1.3 Factoring: -27y3+27y2-9y+2
Thoughtfully split the expression at hand into groups, each group having two terms :
Group 1: -9y+2
Group 2: -27y3+27y2
Pull out from each group separately :
Group 1: (-9y+2) • (1) = (9y-2) • (-1)
Group 2: (y-1) • (-27y2)
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 :
1.4 Find roots (zeroes) of : F(y) = -27y3+27y2-9y+2
Polynomial Roots Calculator is a set of methods aimed at finding values of y for which F(y)=0
Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers y 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 -27 and the Trailing Constant is 2.
The factor(s) are:
of the Leading Coefficient : 1,3 ,9 ,27
of the Trailing Constant : 1 ,2
Let us test ....
| P | Q | P/Q | F(P/Q) | Divisor | |||||
|---|---|---|---|---|---|---|---|---|---|
| -1 | 1 | -1.00 | 65.00 | ||||||
| -1 | 3 | -0.33 | 9.00 | ||||||
| -1 | 9 | -0.11 | 3.37 | ||||||
| -1 | 27 | -0.04 | 2.37 | ||||||
| -2 | 1 | -2.00 | 344.00 | ||||||
| -2 | 3 | -0.67 | 28.00 | ||||||
| -2 | 9 | -0.22 | 5.63 | ||||||
| -2 | 27 | -0.07 | 2.83 | ||||||
| 1 | 1 | 1.00 | -7.00 | ||||||
| 1 | 3 | 0.33 | 1.00 | ||||||
| 1 | 9 | 0.11 | 1.30 | ||||||
| 1 | 27 | 0.04 | 1.70 | ||||||
| 2 | 1 | 2.00 | -124.00 | ||||||
| 2 | 3 | 0.67 | 0.00 | 3y-2 | |||||
| 2 | 9 | 0.22 | 1.04 | ||||||
| 2 | 27 | 0.07 | 1.47 |
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
-27y3+27y2-9y+2
can be divided with 3y-2
Polynomial Long Division :
1.5 Polynomial Long Division
Dividing : -27y3+27y2-9y+2
("Dividend")
By : 3y-2 ("Divisor")
| dividend | - | 27y3 | + | 27y2 | - | 9y | + | 2 | |
| - divisor | * -9y2 | - | 27y3 | + | 18y2 | ||||
| remainder | 9y2 | - | 9y | + | 2 | ||||
| - divisor | * 3y1 | 9y2 | - | 6y | |||||
| remainder | - | 3y | + | 2 | |||||
| - divisor | * -y0 | - | 3y | + | 2 | ||||
| remainder | 0 |
Quotient : -9y2+3y-1 Remainder: 0
Trying to factor by splitting the middle term
1.6 Factoring 9y2-3y+1
The first term is, 9y2 its coefficient is 9 .
The middle term is, -3y its coefficient is -3 .
The last term, "the constant", is +1
Step-1 : Multiply the coefficient of the first term by the constant 9 • 1 = 9
Step-2 : Find two factors of 9 whose sum equals the coefficient of the middle term, which is -3 .
| -9 | + | -1 | = | -10 | ||
| -3 | + | -3 | = | -6 | ||
| -1 | + | -9 | = | -10 | ||
| 1 | + | 9 | = | 10 | ||
| 3 | + | 3 | = | 6 | ||
| 9 | + | 1 | = | 10 |
Observation : No two such factors can be found !!
Conclusion : Trinomial can not be factored
Final result :
(-9y2 + 3y - 1) • (3y - 2)
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