How do I find non-decimal solutions for the two following cubic equations ?
x^3-11x^2-4x-44=0
x^3+11x^2-4x+44=0
Your Eqs should be:
x^3 + 11x^2 - 4x - 44 = 0.
x^3 - 11x^2 - 4x + 44 = 0.
Add the Eqs:
2x^3 - 8x = 0, (sum).
x^3 - 4x = 0,
x(x^2 - 4) = 0,
x^2 - 4 = 0,
x^2 = 4,
X = 2 and -2.
3rd order (cubic) equations should have 3 solutions
1st eqn ... factoring ... (x + 11)(x^2 - 4) = 0 ... x = -11 , ±2
2nd eqn ... factoring ... (x - 11)(x^2 - 4) = 0 ... x = 11 , ±2
To find non-decimal solutions for the two given cubic equations, you can use various methods such as factoring, graphing, or numerical methods like the Newton-Raphson method. In this case, we will use the Rational Root Theorem to determine if any rational roots exist, and then use synthetic division to find these roots.
For the first equation, x^3 - 11x^2 - 4x - 44 = 0:
Step 1: Apply the Rational Root Theorem.
The Rational Root Theorem states that any rational root of the form p/q, where p is a factor of the constant term (in this case, 44) and q is a factor of the leading coefficient (in this case, 1), may be a solution to the equation. So the possible rational roots are ±(1, 2, 4, 11, 22, 44).
Step 2: Use synthetic division to test each possible rational root.
By using synthetic division with the possible rational roots, we can determine if any of them are actual roots of the equation. Starting with the first possible root, let's say x = 1:
1 │ 1 -11 -4 -44
└─── 1 -10 -14
The remainder is not zero, so x = 1 is not a root.
Step 3: Repeat step 2 with the remaining possible roots.
Next, we try x = -1:
-1 │ 1 -11 -4 -44
└─── -1 12 -8
The remainder is not zero, so x = -1 is not a root.
Step 4: Continue applying synthetic division with the remaining possible roots until a root is found.
Next, we try x = 2:
2 │ 1 -11 -4 -44
└─── 2 -18 -44
The remainder is zero, which means x = 2 is a root of the equation.
Thus, one non-decimal solution of the equation x^3 - 11x^2 - 4x - 44 = 0 is x = 2.
To find additional solutions, we can use polynomial long division to divide the original equation by (x - 2). This will give us a quadratic equation that can be solved using factoring or the quadratic formula.
Now, let's solve the second equation, x^3 +11x^2 - 4x + 44 = 0, following similar steps:
Step 1: Apply the Rational Root Theorem.
The possible rational roots are the same as in the previous equation: ±(1, 2, 4, 11, 22, 44).
Step 2: Use synthetic division to test each possible rational root.
Let's first try x = 1:
1 │ 1 11 -4 44
└─── 1 12 8
The remainder is not zero, so x = 1 is not a root.
Step 3: Repeat step 2 with the remaining possible roots.
Next, we try x = -1:
-1 │ 1 11 -4 44
└─── -1 0 4
The remainder is not zero, so x = -1 is not a root.
Step 4: Continue applying synthetic division with the remaining possible roots until a root is found.
Next, we try x = 2:
2 │ 1 11 -4 44
└─── 2 26 44
The remainder is not zero, so x = 2 is not a root.
Step 5: Repeat the process until a root is found.
Next, we try x = -2:
-2 │ 1 11 -4 44
└─── -2 6 -20
The remainder is not zero, so x = -2 is not a root.
Step 6: Repeat the process until a root is found.
Next, we try x = 4:
4 │ 1 11 -4 44
└─── 4 60 224
The remainder is not zero, so x = 4 is not a root.
Step 7: Repeat the process until a root is found.
Next, we try x = -4:
-4 │ 1 11 -4 44
└─── -4 0 -4
The remainder is zero, which means x = -4 is a root of the equation.
So, one non-decimal solution of x^3 + 11x^2 - 4x + 44 = 0 is x = -4.
Again, to find additional solutions, we can use polynomial long division to divide the original equation by (x + 4) and solve the resulting quadratic equation.