Solve the poloynomial
x^3+4x^2-x-4=0
x^3+4x^2-x-4=0
use grouping
x^2(x+4) - (x+4) = 0
(x+4)(x^2 - 1) = 0
(x+4)(x+1)(x-1) = 0
x = -4, ± 1
To solve the given polynomial equation x^3 + 4x^2 - x - 4 = 0, we can use a combination of factoring and the Rational Root Theorem.
Step 1: Factor out any common factors, if possible.
In this equation, there are no common factors to be factorized.
Step 2: Apply the Rational Root Theorem.
The Rational Root Theorem states that any rational root of a polynomial equation of the form ax^n + bx^(n-1) + ... + kx + l = 0 must be of the form p/q, where p is a factor of the constant term l and q is a factor of the leading coefficient a.
So, for our given equation, the leading coefficient (a) is 1 and the constant term (l) is -4. Therefore, the possible rational roots can be +4, -4, +1, -1.
Step 3: Test the possible roots using synthetic division.
We can use synthetic division to test each of the possible roots and see if they give a remainder of zero when substituted into the equation.
Let's test the first possible root, x = +4:
Using synthetic division, we setup the division as follows:
4 | 1 4 -1 -4
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| 4 32 124
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Since the remainder is not zero, x = +4 is not a root of the polynomial.
Similarly, we can test the other three possible rational roots: x = -4, x = +1, and x = -1.
Testing x = -4:
-4 | 1 4 -1 -4
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| -4 0 4
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Since the remainder is zero, x = -4 is a root of the polynomial.
Step 4: Find the remaining quadratic equation.
After identifying one root, which is x = -4, we can now divide the original cubic polynomial by (x + 4) using polynomial long division or synthetic division.
Using synthetic division:
-4 | 1 4 -1 -4
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| -4 0 4
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1 0 -1 0
The result is a quadratic equation: x^2 - 1 = 0.
Step 5: Solve the quadratic equation.
To solve the quadratic equation x^2 - 1 = 0, we can factor it as (x + 1)(x - 1) = 0.
Setting each factor equal to zero, we get:
x + 1 = 0 or x - 1 = 0
Solving for x in each case, we have:
x = -1 or x = 1
Step 6: Find all the roots.
The roots of the given cubic polynomial equation x^3 + 4x^2 - x - 4 = 0 are x = -4, x = -1, and x = 1.
Therefore, the solutions to the equation are x = -4, x = -1, and x = 1.