solve (x^3)+(12x^2)+(21x)-(4)=(x^4)-(2x^3)-(13x^2)-(4)
the answers are x=-1,-3,7,0 but i need to know how to do it
x^4 -3x^3 -25 x^2 -21 x = 0
x ( x^3 -3 x^2 -25 x -21) = 0
so we know x = 0 is a solution
Guess -1 solution
(You can see that x^3 -3 x^2 -25 x -21 is 0 for x = -1)
then we know (x+1) is a factor of
x^3 -3 x^2 -25 x -21
so do long division
(x^3 -3 x^2 -25 x -21)/(x+1) =
x^2-4x-21
(x+3)(x-7) = 0
so
x = -3 or +7
By the way, you could guess the seven and three immediately from the 21 but I saw the -1 first.
To solve the given equation:
1. Start by setting the equation equal to zero:
(x^4) - (2x^3) - (13x^2) - (4) - (x^3) - (12x^2) - (21x) + (4) = 0
2. Combine like terms:
(x^4 - 2x^3 - x^3) + (13x^2 - 12x^2 - 21x) + (4 - 4) = 0
3. Simplify the equation:
x^4 - 3x^3 - 12x^2 - 21x = 0
4. Factor out an "x" from the equation:
x(x^3 - 3x^2 - 12x - 21) = 0
5. Set each factor equal to zero and solve for "x":
a) x = 0
b) x^3 - 3x^2 - 12x - 21 = 0
6. To solve the cubic equation x^3 - 3x^2 - 12x - 21 = 0, you can use various methods such as graphing, factoring, or using synthetic division to find the roots.
In this case, let's use synthetic division to find the rational roots:
- Start by assuming possible rational roots:
Since the constant term is -21, the possible rational roots can be obtained by applying the Rational Root Theorem. The possible rational roots are: ±1, ±3, ±7, ±21.
- Use synthetic division with the possible rational roots to test for actual roots:
Synthetic division is a method used to perform long division quickly and efficiently. Let's try the possible rational roots one by one until we find a root that satisfies the equation.
- Testing x = -1:
Using synthetic division, we have:
-1 | 1 -3 -12 -21
|__ 1____2___10______
The remainder is 0, which means x = -1 is a root.
So far, we have found one root: x = -1.
- Testing x = 1, 3, and so on, until we find all the roots.
Continuing the process, we find the following roots after testing all possible rational roots:
x = -1, -3, 7, 0
7. Therefore, the solutions to the given equation are: x = -1, -3, 7, 0.