Use rational root theorem and the factor theorem to help solve the following equation
X4-2x3-13x2+38x-24=0
If there rational roots to
x^4 - 2x^3 - 13x^2 + 38x - 24 = 0
then the numerator must divide 24 and the denominator must divide 1.
In other words, the roots must be a factor of 24, in this case.
The factor theorem says that if x-a divides f(x), then a is a root of f(x) = 0.
A little easy synthetic division reveals that roots are present at
x = 1,2,3,-4
To solve the equation X^4 - 2x^3 - 13x^2 + 38x - 24 = 0 using the rational root theorem and the factor theorem, we can follow these steps:
Step 1: Apply the rational root theorem.
The rational root theorem states that if a polynomial equation has a rational root 𝑟, then 𝑟 must be of the form 𝑟 = 𝑝/𝑞, where 𝑝 is a factor of the constant term (in this case, -24) and 𝑞 is a factor of the leading coefficient (in this case, 1).
So, we need to find the factors of -24 and 1:
Factors of -24: ±1, ±2, ±3, ±4, ±6, ±8, ±12, ±24
Factors of 1: ±1
Possible rational roots: ±1, ±2, ±3, ±4, ±6, ±8, ±12, ±24
Step 2: Check the possible roots using synthetic division.
We will check each of the possible rational roots using synthetic division to see if they are indeed roots of the equation.
Checking for x = 1:
1 | 1 -2 -13 38 -24
| 1 -1 -14 24
|---------------------------------
| 1 -1 -14 24 0
Checking for x = -1:
-1 | 1 -2 -13 38 -24
| -1 3 10 -28
|---------------------------------
| 1 -3 -10 48 -52
Checking for x = 2:
2 | 1 -2 -13 38 -24
| 2 0 -26 24
|---------------------------------
| 1 0 -13 12 0
Checking for x = -2:
-2 | 1 -2 -13 38 -24
| -2 8 -10 24
|---------------------------------
| 1 -4 -5 28 0
Checking for x = 3:
3 | 1 -2 -13 38 -24
| 3 3 -30 24
|---------------------------------
| 1 1 -10 8 0
Checking for x = -3:
-3 | 1 -2 -13 38 -24
| -3 9 -12 12
|---------------------------------
| 1 -5 -4 26 0
Checking for x = 4:
4 | 1 -2 -13 38 -24
| 4 8 -20 72
|---------------------------------
| 1 2 -5 18 48
Checking for x = -4:
-4 | 1 -2 -13 38 -24
| -4 24 -44 24
|---------------------------------
| 1 -6 11 -6 0
Checking for x = 6:
6 | 1 -2 -13 38 -24
| 6 24 66 252
|---------------------------------
| 1 4 11 104 228
Checking for x = -6:
-6 | 1 -2 -13 38 -24
| -6 48 -210 252
|---------------------------------
| 1 -8 35 -172 228
Checking for x = 8:
8 | 1 -2 -13 38 -24
| 8 48 280 432
|---------------------------------
| 1 6 35 318 408
Checking for x = -8:
-8 | 1 -2 -13 38 -24
| -8 80 -1344 924
|---------------------------------
| 1 -10 67 -1306 900
From the synthetic division, we can see that the only rational root of the equation is x = 4.
Step 3: Applying the factor theorem.
Now that we have found one root, x = 4, we can divide the polynomial equation by (x - 4) using synthetic division to find the remaining roots.
4 | 1 -2 -13 38 -24
| 4 8 -20 72
|---------------------------------
| 1 2 -5 18 48
The quotient is x^3 + 2x^2 - 5x + 18.
We can factor this as:
(x^3 + 2x^2 - 5x + 18) = (x - 1)(x^2 + 3x - 18).
Now, we have x - 1 = 0 or x^2 + 3x - 18 = 0.
Solving x - 1 = 0 gives x = 1.
To solve x^2 + 3x - 18 = 0, we can either factor it or use the quadratic formula.
Factoring x^2 + 3x - 18 = 0, we get:
(x + 6)(x - 3) = 0.
Therefore, the remaining roots are x = -6 and x = 3.
In conclusion, the solutions to the equation X^4 - 2x^3 - 13x^2 + 38x - 24 = 0 are x = 1, x = 3, x = 4, and x = -6.