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 :
((((8•(x4))-(6•(x3)))-(3•5•11x2))-37x)+60Step 2 :
Equation at the end of step 2 :
((((8•(x4))-(2•3x3))-(3•5•11x2))-37x)+60Step 3 :
Equation at the end of step 3 :
(((23x4 - (2•3x3)) - (3•5•11x2)) - 37x) + 60
Step 4 :
Polynomial Roots Calculator :
4.1 Find roots (zeroes) of : F(x) = 8x4-6x3-165x2-37x+60
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 60.
The factor(s) are:
of the Leading Coefficient : 1,2 ,4 ,8
of the Trailing Constant : 1 ,2 ,3 ,4 ,5 ,6 ,10 ,12 ,15 ,20 , etc
Let us test ....
| P | Q | P/Q | F(P/Q) | Divisor | |||||
|---|---|---|---|---|---|---|---|---|---|
| -1 | 1 | -1.00 | -54.00 | ||||||
| -1 | 2 | -0.50 | 38.50 | ||||||
| -1 | 4 | -0.25 | 59.06 | ||||||
| -1 | 8 | -0.12 | 62.06 | ||||||
| -2 | 1 | -2.00 | -350.00 | ||||||
| -3 | 1 | -3.00 | -504.00 | ||||||
| -3 | 2 | -1.50 | -195.00 | ||||||
| -3 | 4 | -0.75 | 0.00 | 4x+3 | |||||
| -3 | 8 | -0.38 | 51.15 | ||||||
| -4 | 1 | -4.00 | 0.00 | x+4 | |||||
| -5 | 1 | -5.00 | 1870.00 | ||||||
| -5 | 2 | -2.50 | -472.50 | ||||||
| -5 | 4 | -1.25 | -120.31 | ||||||
| -5 | 8 | -0.62 | 21.36 | ||||||
| -6 | 1 | -6.00 | 6006.00 | ||||||
| -10 | 1 | -10.00 | 69930.00 | ||||||
| -12 | 1 | -12.00 | 153000.00 | ||||||
| -15 | 1 | -15.00 | 388740.00 | ||||||
| -15 | 2 | -7.50 | 18900.00 | ||||||
| -15 | 4 | -3.75 | -223.12 | ||||||
| -15 | 8 | -1.88 | -312.28 | ||||||
| -20 | 1 | -20.00 | 1262800.00 | ||||||
| 1 | 1 | 1.00 | -140.00 | ||||||
| 1 | 2 | 0.50 | 0.00 | 2x-1 | |||||
| 1 | 4 | 0.25 | 40.38 | ||||||
| 1 | 8 | 0.12 | 52.79 | ||||||
| 2 | 1 | 2.00 | -594.00 | ||||||
| 3 | 1 | 3.00 | -1050.00 | ||||||
| 3 | 2 | 1.50 | -346.50 | ||||||
| 3 | 4 | 0.75 | -60.56 | ||||||
| 3 | 8 | 0.38 | 22.76 | ||||||
| 4 | 1 | 4.00 | -1064.00 | ||||||
| 5 | 1 | 5.00 | 0.00 | x-5 | |||||
| 5 | 2 | 2.50 | -845.00 | ||||||
| 5 | 4 | 1.25 | -236.25 | ||||||
| 5 | 8 | 0.62 | -27.82 | ||||||
| 6 | 1 | 6.00 | 2970.00 | ||||||
| 10 | 1 | 10.00 | 57190.00 | ||||||
| 12 | 1 | 12.00 | 131376.00 | ||||||
| 15 | 1 | 15.00 | 347130.00 | ||||||
| 15 | 2 | 7.50 | 13282.50 | ||||||
| 15 | 4 | 3.75 | -1133.44 | ||||||
| 15 | 8 | 1.88 | -530.13 | ||||||
| 20 | 1 | 20.00 | 1165320.00 |
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
8x4-6x3-165x2-37x+60
can be divided by 4 different polynomials,including by x-5
Polynomial Long Division :
4.2 Polynomial Long Division
Dividing : 8x4-6x3-165x2-37x+60
("Dividend")
By : x-5 ("Divisor")
| dividend | 8x4 | - | 6x3 | - | 165x2 | - | 37x | + | 60 | ||
| - divisor | * 8x3 | 8x4 | - | 40x3 | |||||||
| remainder | 34x3 | - | 165x2 | - | 37x | + | 60 | ||||
| - divisor | * 34x2 | 34x3 | - | 170x2 | |||||||
| remainder | 5x2 | - | 37x | + | 60 | ||||||
| - divisor | * 5x1 | 5x2 | - | 25x | |||||||
| remainder | - | 12x | + | 60 | |||||||
| - divisor | * -12x0 | - | 12x | + | 60 | ||||||
| remainder | 0 |
Quotient : 8x3+34x2+5x-12 Remainder: 0
Polynomial Roots Calculator :
4.3 Find roots (zeroes) of : F(x) = 8x3+34x2+5x-12
See theory in step 4.1
In this case, the Leading Coefficient is 8 and the Trailing Constant is -12.
The factor(s) are:
of the Leading Coefficient : 1,2 ,4 ,8
of the Trailing Constant : 1 ,2 ,3 ,4 ,6 ,12
Let us test ....
| P | Q | P/Q | F(P/Q) | Divisor | |||||
|---|---|---|---|---|---|---|---|---|---|
| -1 | 1 | -1.00 | 9.00 | ||||||
| -1 | 2 | -0.50 | -7.00 | ||||||
| -1 | 4 | -0.25 | -11.25 | ||||||
| -1 | 8 | -0.12 | -12.11 | ||||||
| -2 | 1 | -2.00 | 50.00 | ||||||
| -3 | 1 | -3.00 | 63.00 | ||||||
| -3 | 2 | -1.50 | 30.00 | ||||||
| -3 | 4 | -0.75 | 0.00 | 4x+3 | |||||
| -3 | 8 | -0.38 | -9.52 | ||||||
| -4 | 1 | -4.00 | 0.00 | x+4 | |||||
| -6 | 1 | -6.00 | -546.00 | ||||||
| -12 | 1 | -12.00 | -9000.00 | ||||||
| 1 | 1 | 1.00 | 35.00 | ||||||
| 1 | 2 | 0.50 | 0.00 | 2x-1 | |||||
| 1 | 4 | 0.25 | -8.50 | ||||||
| 1 | 8 | 0.12 | -10.83 | ||||||
| 2 | 1 | 2.00 | 198.00 | ||||||
| 3 | 1 | 3.00 | 525.00 | ||||||
| 3 | 2 | 1.50 | 99.00 | ||||||
| 3 | 4 | 0.75 | 14.25 | ||||||
| 3 | 8 | 0.38 | -4.92 | ||||||
| 4 | 1 | 4.00 | 1064.00 | ||||||
| 6 | 1 | 6.00 | 2970.00 | ||||||
| 12 | 1 | 12.00 | 18768.00 |
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+34x2+5x-12
can be divided by 3 different polynomials,including by 2x-1
Polynomial Long Division :
4.4 Polynomial Long Division
Dividing : 8x3+34x2+5x-12
("Dividend")
By : 2x-1 ("Divisor")
| dividend | 8x3 | + | 34x2 | + | 5x | - | 12 | ||
| - divisor | * 4x2 | 8x3 | - | 4x2 | |||||
| remainder | 38x2 | + | 5x | - | 12 | ||||
| - divisor | * 19x1 | 38x2 | - | 19x | |||||
| remainder | 24x | - | 12 | ||||||
| - divisor | * 12x0 | 24x | - | 12 | |||||
| remainder | 0 |
Quotient : 4x2+19x+12 Remainder: 0
Trying to factor by splitting the middle term
4.5 Factoring 4x2+19x+12
The first term is, 4x2 its coefficient is 4 .
The middle term is, +19x its coefficient is 19 .
The last term, "the constant", is +12
Step-1 : Multiply the coefficient of the first term by the constant 4 • 12 = 48
Step-2 : Find two factors of 48 whose sum equals the coefficient of the middle term, which is 19 .
| -48 | + | -1 | = | -49 | ||
| -24 | + | -2 | = | -26 | ||
| -16 | + | -3 | = | -19 | ||
| -12 | + | -4 | = | -16 | ||
| -8 | + | -6 | = | -14 | ||
| -6 | + | -8 | = | -14 | ||
| -4 | + | -12 | = | -16 | ||
| -3 | + | -16 | = | -19 | ||
| -2 | + | -24 | = | -26 | ||
| -1 | + | -48 | = | -49 | ||
| 1 | + | 48 | = | 49 | ||
| 2 | + | 24 | = | 26 | ||
| 3 | + | 16 | = | 19 | That's it |
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, 3 and 16
4x2 + 3x + 16x + 12
Step-4 : Add up the first 2 terms, pulling out like factors :
x • (4x+3)
Add up the last 2 terms, pulling out common factors :
4 • (4x+3)
Step-5 : Add up the four terms of step 4 :
(x+4) • (4x+3)
Which is the desired factorization
Final result :
(4x + 3) • (x + 4) • (2x - 1) • (x - 5)
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