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 :
(((48•(x4))+(36•(x3)))-(2•3•23x2))-36xStep 2 :
Equation at the end of step 2 :
(((48 • (x4)) + (22•32x3)) - (2•3•23x2)) - 36xStep 3 :
Equation at the end of step 3 :
(((24•3x4) + (22•32x3)) - (2•3•23x2)) - 36x
Step 4 :
Step 5 :
Pulling out like terms :
5.1 Pull out like factors :
48x4 + 36x3 - 138x2 - 36x =
6x • (8x3 + 6x2 - 23x - 6)
Checking for a perfect cube :
5.2 8x3 + 6x2 - 23x - 6 is not a perfect cube
Trying to factor by pulling out :
5.3 Factoring: 8x3 + 6x2 - 23x - 6
Thoughtfully split the expression at hand into groups, each group having two terms :
Group 1: -23x - 6
Group 2: 6x2 + 8x3
Pull out from each group separately :
Group 1: (23x + 6) • (-1)
Group 2: (4x + 3) • (2x2)
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 :
5.4 Find roots (zeroes) of : F(x) = 8x3 + 6x2 - 23x - 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 8 and the Trailing Constant is -6.
The factor(s) are:
of the Leading Coefficient : 1,2 ,4 ,8
of the Trailing Constant : 1 ,2 ,3 ,6
Let us test ....
| P | Q | P/Q | F(P/Q) | Divisor | |||||
|---|---|---|---|---|---|---|---|---|---|
| -1 | 1 | -1.00 | 15.00 | ||||||
| -1 | 2 | -0.50 | 6.00 | ||||||
| -1 | 4 | -0.25 | 0.00 | 4x + 1 | |||||
| -1 | 8 | -0.12 | -3.05 | ||||||
| -2 | 1 | -2.00 | 0.00 | x + 2 | |||||
| -3 | 1 | -3.00 | -99.00 | ||||||
| -3 | 2 | -1.50 | 15.00 | ||||||
| -3 | 4 | -0.75 | 11.25 | ||||||
| -3 | 8 | -0.38 | 3.05 | ||||||
| -6 | 1 | -6.00 | -1380.00 | ||||||
| 1 | 1 | 1.00 | -15.00 | ||||||
| 1 | 2 | 0.50 | -15.00 | ||||||
| 1 | 4 | 0.25 | -11.25 | ||||||
| 1 | 8 | 0.12 | -8.77 | ||||||
| 2 | 1 | 2.00 | 36.00 | ||||||
| 3 | 1 | 3.00 | 195.00 | ||||||
| 3 | 2 | 1.50 | 0.00 | 2x - 3 | |||||
| 3 | 4 | 0.75 | -16.50 | ||||||
| 3 | 8 | 0.38 | -13.36 | ||||||
| 6 | 1 | 6.00 | 1800.00 |
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
8x3 + 6x2 - 23x - 6
can be divided by 3 different polynomials,including by 2x - 3
Polynomial Long Division :
5.5 Polynomial Long Division
Dividing : 8x3 + 6x2 - 23x - 6
("Dividend")
By : 2x - 3 ("Divisor")
| dividend | 8x3 | + | 6x2 | - | 23x | - | 6 | ||
| - divisor | * 4x2 | 8x3 | - | 12x2 | |||||
| remainder | 18x2 | - | 23x | - | 6 | ||||
| - divisor | * 9x1 | 18x2 | - | 27x | |||||
| remainder | 4x | - | 6 | ||||||
| - divisor | * 2x0 | 4x | - | 6 | |||||
| remainder | 0 |
Quotient : 4x2+9x+2 Remainder: 0
Trying to factor by splitting the middle term
5.6 Factoring 4x2+9x+2
The first term is, 4x2 its coefficient is 4 .
The middle term is, +9x its coefficient is 9 .
The last term, "the constant", is +2
Step-1 : Multiply the coefficient of the first term by the constant 4 • 2 = 8
Step-2 : Find two factors of 8 whose sum equals the coefficient of the middle term, which is 9 .
| -8 | + | -1 | = | -9 | ||
| -4 | + | -2 | = | -6 | ||
| -2 | + | -4 | = | -6 | ||
| -1 | + | -8 | = | -9 | ||
| 1 | + | 8 | = | 9 | That's it |
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, 1 and 8
4x2 + 1x + 8x + 2
Step-4 : Add up the first 2 terms, pulling out like factors :
x • (4x+1)
Add up the last 2 terms, pulling out common factors :
2 • (4x+1)
Step-5 : Add up the four terms of step 4 :
(x+2) • (4x+1)
Which is the desired factorization
Final result :
6x • (4x + 1) • (x + 2) • (2x - 3)
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