Finding The Roots Of Polynomials - Tiger Algebra

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Step 1 :

Equation at the end of step 1 :

((((9•(x4))-(9•(x3)))-(2•29x2))+4x)+24

Step 2 :

Equation at the end of step 2 :

((((9•(x4))-32x3)-(2•29x2))+4x)+24

Step 3 :

Equation at the end of step 3 :

(((32x4 - 32x3) - (2•29x2)) + 4x) + 24

Step 4 :

Polynomial Roots Calculator :

4.1 Find roots (zeroes) of : F(x) = 9x4-9x3-58x2+4x+24Polynomial 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 integersThe 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 CoefficientIn this case, the Leading Coefficient is 9 and the Trailing Constant is 24. The factor(s) are: of the Leading Coefficient : 1,3 ,9 of the Trailing Constant : 1 ,2 ,3 ,4 ,6 ,8 ,12 ,24 Let us test ....

PQP/QF(P/Q)Divisor
-1 1 -1.00 -20.00
-1 3 -0.33 16.67
-1 9 -0.11 22.85
-2 1 -2.00 0.00 x+2
-2 3 -0.67 0.00 3x+2
-2 9 -0.22 20.37
-3 1 -3.00 462.00
-4 1 -4.00 1960.00
-4 3 -1.33 -34.67
-4 9 -0.44 11.91
-6 1 -6.00 11520.00
-8 1 -8.00 37752.00
-8 3 -2.67 226.67
-8 9 -0.89 -13.44
-12 1 -12.00 193800.00
-24 1 -24.00 3076920.00
1 1 1.00 -30.00
1 3 0.33 18.67
1 9 0.11 23.72
2 1 2.00 -128.00
2 3 0.67 0.00 3x-2
2 9 0.22 21.95
3 1 3.00 0.00 x-3
4 1 4.00 840.00
4 3 1.33 -66.67
4 9 0.44 13.88
6 1 6.00 7680.00
8 1 8.00 28600.00
8 3 2.67 -93.33
8 9 0.89 -18.97
12 1 12.00 162792.00
24 1 24.00 2828280.00

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 9x4-9x3-58x2+4x+24 can be divided by 4 different polynomials,including by x-3

Polynomial Long Division :

4.2 Polynomial Long Division Dividing : 9x4-9x3-58x2+4x+24 ("Dividend") By : x-3 ("Divisor")

dividend 9x4 - 9x3 - 58x2 + 4x + 24
- divisor * 9x3 9x4 - 27x3
remainder 18x3 - 58x2 + 4x + 24
- divisor * 18x2 18x3 - 54x2
remainder- 4x2 + 4x + 24
- divisor * -4x1 - 4x2 + 12x
remainder- 8x + 24
- divisor * -8x0 - 8x + 24
remainder0

Quotient : 9x3+18x2-4x-8 Remainder: 0

Polynomial Roots Calculator :

4.3 Find roots (zeroes) of : F(x) = 9x3+18x2-4x-8 See theory in step 4.1 In this case, the Leading Coefficient is 9 and the Trailing Constant is -8. The factor(s) are: of the Leading Coefficient : 1,3 ,9 of the Trailing Constant : 1 ,2 ,4 ,8 Let us test ....

PQP/QF(P/Q)Divisor
-1 1 -1.00 5.00
-1 3 -0.33 -5.00
-1 9 -0.11 -7.35
-2 1 -2.00 0.00 x+2
-2 3 -0.67 0.00 3x+2
-2 9 -0.22 -6.32
-4 1 -4.00 -280.00
-4 3 -1.33 8.00
-4 9 -0.44 -3.46
-8 1 -8.00 -3432.00
-8 3 -2.67 -40.00
-8 9 -0.89 3.46
1 1 1.00 15.00
1 3 0.33 -7.00
1 9 0.11 -8.21
2 1 2.00 128.00
2 3 0.67 0.00 3x-2
2 9 0.22 -7.90
4 1 4.00 840.00
4 3 1.33 40.00
4 9 0.44 -5.43
8 1 8.00 5720.00
8 3 2.67 280.00
8 9 0.89 8.99

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 9x3+18x2-4x-8 can be divided by 3 different polynomials,including by 3x-2

Polynomial Long Division :

4.4 Polynomial Long Division Dividing : 9x3+18x2-4x-8 ("Dividend") By : 3x-2 ("Divisor")

dividend 9x3 + 18x2 - 4x - 8
- divisor * 3x2 9x3 - 6x2
remainder 24x2 - 4x - 8
- divisor * 8x1 24x2 - 16x
remainder 12x - 8
- divisor * 4x0 12x - 8
remainder0

Quotient : 3x2+8x+4 Remainder: 0

Trying to factor by splitting the middle term

4.5 Factoring 3x2+8x+4 The first term is, 3x2 its coefficient is 3 .The middle term is, +8x its coefficient is 8 .The last term, "the constant", is +4 Step-1 : Multiply the coefficient of the first term by the constant 3 • 4 = 12 Step-2 : Find two factors of 12 whose sum equals the coefficient of the middle term, which is 8 .

-12 + -1 = -13
-6 + -2 = -8
-4 + -3 = -7
-3 + -4 = -7
-2 + -6 = -8
-1 + -12 = -13
1 + 12 = 13
2 + 6 = 8 That's it

Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, 2 and 6 3x2 + 2x + 6x + 4Step-4 : Add up the first 2 terms, pulling out like factors : x • (3x+2) Add up the last 2 terms, pulling out common factors : 2 • (3x+2) Step-5 : Add up the four terms of step 4 : (x+2) • (3x+2) Which is the desired factorization

Final result :

(3x + 2) • (x + 2) • (3x - 2) • (x - 3)

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