Find roots for 2x3 - 4x2 - 22x + 24
Rational roots of a polynomial will be q/p where
q is a factor of the constant term (24)
and p is a factor of the leading x3 coefficient (2)
Determine our list of p values first:
| Numbers (1 - 2) | 2 ÷ Number List | Factor of p? |
|---|---|---|
| 1 | 2 ÷ 1 = 2 | Y |
| 2 | 2 ÷ 2 = 1 | Y |
Let's determine our list of q values next:
| Numbers (1 - 24) | 24 ÷ Number List | Factor of q? |
|---|---|---|
| 1 | 24 ÷ 1 = 24 | Y |
| 2 | 24 ÷ 2 = 12 | Y |
| 3 | 24 ÷ 3 = 8 | Y |
| 4 | 24 ÷ 4 = 6 | Y |
| 6 | 24 ÷ 6 = 4 | Y |
| 8 | 24 ÷ 8 = 3 | Y |
| 12 | 24 ÷ 12 = 2 | Y |
| 24 | 24 ÷ 24 = 1 | Y |
Calculate our q ÷ p = r values
| p | q | r = q ÷ p | ƒ(r) = 2r3 - 4r2 - 22r + 24 | ƒ(r) value | -1 x r | ƒ(-r) = 2r3 - 4r2 - 22r + 24 | ƒ(-r) value |
|---|---|---|---|---|---|---|---|
| 1 | 1 | 1 | 2(1)3 - 4(1)2 - 22(1) + 24 | 0 | -1 | 2(-1)3 - 4(-1)2 - 22(-1) + 24 | 40 |
| 1 | 2 | 2 | 2(2)3 - 4(2)2 - 22(2) + 24 | -20 | -2 | 2(-2)3 - 4(-2)2 - 22(-2) + 24 | 36 |
| 1 | 3 | 3 | 2(3)3 - 4(3)2 - 22(3) + 24 | -24 | -3 | 2(-3)3 - 4(-3)2 - 22(-3) + 24 | 0 |
| 1 | 4 | 4 | 2(4)3 - 4(4)2 - 22(4) + 24 | 0 | -4 | 2(-4)3 - 4(-4)2 - 22(-4) + 24 | -80 |
| 1 | 6 | 6 | 2(6)3 - 4(6)2 - 22(6) + 24 | 180 | -6 | 2(-6)3 - 4(-6)2 - 22(-6) + 24 | -420 |
| 1 | 8 | 8 | 2(8)3 - 4(8)2 - 22(8) + 24 | 616 | -8 | 2(-8)3 - 4(-8)2 - 22(-8) + 24 | -1080 |
| 1 | 12 | 12 | 2(12)3 - 4(12)2 - 22(12) + 24 | 2640 | -12 | 2(-12)3 - 4(-12)2 - 22(-12) + 24 | -3744 |
| 1 | 24 | 24 | 2(24)3 - 4(24)2 - 22(24) + 24 | 24840 | -24 | 2(-24)3 - 4(-24)2 - 22(-24) + 24 | -29400 |
| 2 | 1 | 0.5 | 2(0.5)3 - 4(0.5)2 - 22(0.5) + 24 | 12.25 | -0.5 | 2(-0.5)3 - 4(-0.5)2 - 22(-0.5) + 24 | 33.75 |
| 2 | 3 | 1.5 | 2(1.5)3 - 4(1.5)2 - 22(1.5) + 24 | -11.25 | -1.5 | 2(-1.5)3 - 4(-1.5)2 - 22(-1.5) + 24 | 41.25 |
Real Roots → ƒ(r) = 0
Root List = {1,-3,4}These are the root(s) using direct substitution.
Below is a link using synthetic division
Click here to see the synthetic division for our polynomial using our root of 1
Click here to see the synthetic division for our polynomial using our root of -3
Click here to see the synthetic division for our polynomial using our root of 4
Final Answer
(4, -3, 1)
You have 2 free calculationss remaining
How does the Cubic Equation Calculator work?
Free Cubic Equation Calculator - Solves for cubic equations in the form ax3 + bx2 + cx + d = 0 using the following methods:
1) Solve the long way for all 3 roots and the discriminant Δ
2) Rational Root Theorem (Rational Zero Theorem) to solve for real roots followed by the synthetic div/quadratic method for the other imaginary roots if applicable.
This calculator has 5 inputs.
1) Solve the long way for all 3 roots and the discriminant Δ
2) Rational Root Theorem (Rational Zero Theorem) to solve for real roots followed by the synthetic div/quadratic method for the other imaginary roots if applicable.
This calculator has 5 inputs.
What 4 formulas are used for the Cubic Equation Calculator?
Δ = 4b3d - b2c2 + 4ac3 - 18abcd + 27a2d2
x1 = 2j * Cosine(k/3) - b/(3a)
x2 = l(m + n) + p
x3 = l(m - n) + p
For more math formulas, check out our Formula Dossier
x1 = 2j * Cosine(k/3) - b/(3a)
x2 = l(m + n) + p
x3 = l(m - n) + p
For more math formulas, check out our Formula Dossier
What 7 concepts are covered in the Cubic Equation Calculator?
- cubic
- cubic equation
- An equation of the form ax3 + bx2 + cx + d = 0
- equation
- a statement declaring two mathematical expressions are equal
- quadratic
- Polynomials with a maximum term degree as the second degree
- rational root theorem
- used to find the rational solutions of a polynomial equation
- synthetic division
- a shorthand method for dividing a polynomial by a linear factor
- unknown
- a number or value we do not know
Example calculations for the Cubic Equation Calculator
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