How do you factor a quartic equation such as 4x^3-24x^2-x+6. Could you break it down step by step? thank you.
This is an example of "gouping"
4x^3-24x^2-x+6
= 4x^2(x-6) - (x-6)
= (x-6)(x^2 - 1) , now by difference of squares
= (x-6)(x-1)(x+1)
5r-5s
------
5r+ 5s
To factor a quartic equation like 4x^3-24x^2-x+6, you can use a combination of factoring by grouping and the rational root theorem. Here are the steps to factor this quartic equation:
Step 1: Check for common factors
Check if there are any common factors among the terms of the equation. In this case, there are no common factors.
Step 2: Find the possible rational roots
Use the rational root theorem to find the possible rational roots of the equation. The rational root theorem states that any rational root of the equation should be of the form p/q, where p is a factor of the constant term (in this case, 6) and q is a factor of the leading coefficient (in this case, 4). The factors of 6 are ±1, ±2, ±3, and ±6, while the factors of 4 are ±1 and ±2. So, the possible rational roots are ±1, ±2, ±3, ±6.
Step 3: Test the possible rational roots using synthetic division
Perform synthetic division to test each of the possible rational roots. Synthetic division helps us determine if a specific root is a zero of the equation. Start by setting up a synthetic division table with the coefficients of the equation.
| 4 -24 -1 6
-----------------------------
1 |
Now, divide the first coefficient (4) by the possible root (1) to get the result:
| 4 -24 -1 6
-----------------------------
1 | 4
-----
Now, multiply the possible root (1) by the result:
| 4 -24 -1 6
-----------------------------
1 | 4 | 4
------
(add)
Repeat the process with the second coefficient:
| 4 -24 -1 6
-----------------------------
1 | 4 | 4 |
------ + 4
-20
Continue the process until you reach the last coefficient:
| 4 -24 -1 6
-----------------------------
1 | 4 | 4 | 3 | 9 |
------ + -1 |
-20 -6 +
4 3
The final result represents the coefficients of the quotient polynomial after dividing by the possible root. In this case, the quotient polynomial is 4x^2 + 4x + 3, and the remainder is 9.
Step 4: Repeat step 3 for all possible rational roots
Repeat the synthetic division process for each of the possible rational roots, ±1, ±2, ±3, and ±6. If a possible root gives a remainder of 0, it means it is a zero of the equation, and you can proceed to the next step.
Step 5: Factor the polynomial
After finding the zeros of the equation, you can express the polynomial as a product of its factors. In this case, the factors will be (x - root1)(x - root2)(x - root3)..., where root1, root2, root3, etc., correspond to the zeros.
For example, if the zeros found in step 4 are root1 = 2, root2 = -1, and root3 = 3, the factors will be (x - 2)(x + 1)(x - 3).
So, the factored form of the quartic equation 4x^3-24x^2-x+6 is (x - 2)(x + 1)(x - 3).