A partial solution set is given for the polynomial equation. Find the complete solution set. (Enter your answers as a comma-separated list.)
x^3 + 3x^2 − 14x − 20 = 0; {−5}
x^3 + 3x^2 − 14x − 20 = (x+5)(x^2-2x-4)
Now just solve the quadratic for the other two values of x.
To find the complete solution set, we need to factorize the polynomial equation. We know that the partial solution given is x = -5, so (x + 5) is a factor of the polynomial.
We can use synthetic division to find the other factor and solve for the remaining solutions:
-5 | 1 3 -14 -20
| -5 10 -20
--------------------
1 -2 -4 -40
The resulting quotient is x^2 - 2x - 4. Now we need to factor this quadratic equation.
To factor x^2 - 2x - 4, we can apply the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
In this case, a = 1, b = -2, and c = -4. Plugging these values into the formula:
x = (-(-2) ± √((-2)^2 - 4(1)(-4))) / (2(1))
= (2 ± √(4 + 16)) / 2
= (2 ± √20) / 2
= (2 ± 2√5) / 2
= 1 ± √5
So the complete solution set is {−5, 1 - √5, 1 + √5}.
To find the complete solution set for the given polynomial equation, we can start by factoring the equation using the known root, which is -5. Here's how we can do it:
Step 1: Divide the polynomial equation by (x - (-5)) or (x + 5) using polynomial long division or synthetic division. By performing the division, we obtain the quotient and remainder.
(x^3 + 3x^2 − 14x − 20) ÷ (x + 5)
Step 2: The quotient will be a quadratic equation, and if there is no remainder (remainder = 0), it means that (x + 5) is a factor of the original equation.
Step 3: Solve the quadratic equation to find the remaining solutions.
Let's go through the steps:
Step 1: Perform the division and determine the quotient:
-5 │ x^3 + 3x^2 - 14x - 20
- (x^3 + 5x^2)
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-2x^2 - 14x
- (-2x^2 - 10x)
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-4x - 20
- (-4x - 20)
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0
The quotient is -2x^2 - 4x.
Step 2: Since the remainder is zero, we can conclude that (x + 5) is a factor of the original equation.
Step 3: Solve the quadratic equation -2x^2 - 4x = 0.
-2x^2 - 4x = 0
-2x(x + 2) = 0
To find the solutions, we set each factor equal to zero:
-2x = 0 x + 2 = 0
x = 0 x = -2
Combining all the solutions, the complete solution set for the polynomial equation x^3 + 3x^2 − 14x − 20 = 0 is {-5, 0, -2}.