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•(x4))+(11•(x3)))-(23•5x2))-132x)+48Step 2 :
Equation at the end of step 2 :
((((3•(x4))+11x3)-(23•5x2))-132x)+48Step 3 :
Equation at the end of step 3 :
(((3x4 + 11x3) - (23•5x2)) - 132x) + 48
Step 4 :
Polynomial Roots Calculator :
4.1 Find roots (zeroes) of : F(x) = 3x4+11x3-40x2-132x+48
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 48.
The factor(s) are:
of the Leading Coefficient : 1,3
of the Trailing Constant : 1 ,2 ,3 ,4 ,6 ,8 ,12 ,16 ,24 ,48
Let us test ....
| P | Q | P/Q | F(P/Q) | Divisor | |||||
|---|---|---|---|---|---|---|---|---|---|
| -1 | 1 | -1.00 | 132.00 | ||||||
| -1 | 3 | -0.33 | 87.19 | ||||||
| -2 | 1 | -2.00 | 112.00 | ||||||
| -2 | 3 | -0.67 | 115.56 | ||||||
| -3 | 1 | -3.00 | 30.00 | ||||||
| -4 | 1 | -4.00 | 0.00 | x+4 | |||||
| -4 | 3 | -1.33 | 136.30 | ||||||
| -6 | 1 | -6.00 | 912.00 | ||||||
| -8 | 1 | -8.00 | 5200.00 | ||||||
| -8 | 3 | -2.67 | 58.67 | ||||||
| -12 | 1 | -12.00 | 39072.00 | ||||||
| -16 | 1 | -16.00 | 143472.00 | ||||||
| -16 | 3 | -5.33 | 372.74 | ||||||
| -24 | 1 | -24.00 | 823440.00 | ||||||
| -48 | 1 | -48.00 | 14622960.00 | ||||||
| 1 | 1 | 1.00 | -110.00 | ||||||
| 1 | 3 | 0.33 | 0.00 | 3x-1 | |||||
| 2 | 1 | 2.00 | -240.00 | ||||||
| 2 | 3 | 0.67 | -53.93 | ||||||
| 3 | 1 | 3.00 | -168.00 | ||||||
| 4 | 1 | 4.00 | 352.00 | ||||||
| 4 | 3 | 1.33 | -163.56 | ||||||
| 6 | 1 | 6.00 | 4080.00 | ||||||
| 8 | 1 | 8.00 | 14352.00 | ||||||
| 8 | 3 | 2.67 | -228.15 | ||||||
| 12 | 1 | 12.00 | 73920.00 | ||||||
| 16 | 1 | 16.00 | 229360.00 | ||||||
| 16 | 3 | 5.33 | 2302.22 | ||||||
| 24 | 1 | 24.00 | 1121232.00 | ||||||
| 48 | 1 | 48.00 | 17043312.00 |
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
3x4+11x3-40x2-132x+48
can be divided by 2 different polynomials,including by 3x-1
Polynomial Long Division :
4.2 Polynomial Long Division
Dividing : 3x4+11x3-40x2-132x+48
("Dividend")
By : 3x-1 ("Divisor")
| dividend | 3x4 | + | 11x3 | - | 40x2 | - | 132x | + | 48 | ||
| - divisor | * x3 | 3x4 | - | x3 | |||||||
| remainder | 12x3 | - | 40x2 | - | 132x | + | 48 | ||||
| - divisor | * 4x2 | 12x3 | - | 4x2 | |||||||
| remainder | - | 36x2 | - | 132x | + | 48 | |||||
| - divisor | * -12x1 | - | 36x2 | + | 12x | ||||||
| remainder | - | 144x | + | 48 | |||||||
| - divisor | * -48x0 | - | 144x | + | 48 | ||||||
| remainder | 0 |
Quotient : x3+4x2-12x-48 Remainder: 0
Polynomial Roots Calculator :
4.3 Find roots (zeroes) of : F(x) = x3+4x2-12x-48
See theory in step 4.1
In this case, the Leading Coefficient is 1 and the Trailing Constant is -48.
The factor(s) are:
of the Leading Coefficient : 1
of the Trailing Constant : 1 ,2 ,3 ,4 ,6 ,8 ,12 ,16 ,24 ,48
Let us test ....
| P | Q | P/Q | F(P/Q) | Divisor | |||||
|---|---|---|---|---|---|---|---|---|---|
| -1 | 1 | -1.00 | -33.00 | ||||||
| -2 | 1 | -2.00 | -16.00 | ||||||
| -3 | 1 | -3.00 | -3.00 | ||||||
| -4 | 1 | -4.00 | 0.00 | x+4 | |||||
| -6 | 1 | -6.00 | -48.00 |
Note - For tidiness, printing of 15 checks which found no root was suppressed
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+4x2-12x-48
can be divided with x+4
Polynomial Long Division :
4.4 Polynomial Long Division
Dividing : x3+4x2-12x-48
("Dividend")
By : x+4 ("Divisor")
| dividend | x3 | + | 4x2 | - | 12x | - | 48 | ||
| - divisor | * x2 | x3 | + | 4x2 | |||||
| remainder | - | 12x | - | 48 | |||||
| - divisor | * 0x1 | ||||||||
| remainder | - | 12x | - | 48 | |||||
| - divisor | * -12x0 | - | 12x | - | 48 | ||||
| remainder | 0 |
Quotient : x2-12 Remainder: 0
Trying to factor as a Difference of Squares :
4.5 Factoring: x2-12
Theory : A difference of two perfect squares, A2 - B2 can be factored into (A+B) • (A-B)
Proof : (A+B) • (A-B) =
A2 - AB + BA - B2 =
A2 - AB + AB - B2 =
A2 - B2
Note : AB = BA is the commutative property of multiplication.
Note : - AB + AB equals zero and is therefore eliminated from the expression.
Check : 12 is not a square !!
Ruling : Binomial can not be factored as the difference of two perfect squares.
Final result :
(x2 - 12) • (x + 4) • (3x - 1)
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