Question Solve the polynomial equation by factoring and then find the solution set. $x^{3} + 8x^{2} = 40x + 08$ . Select the correct choice: A. The solution set is ( ) or B. There is no solution.

Studdy Solution

STEP 1

Assumptions
1. We are given a polynomial equation to solve: $x^{3} + 8x^{2} = 40x + 320$ .
2. We need to factor the polynomial to find the solution set.
3. We will use algebraic manipulation to factor the polynomial.
4. We will check for possible errors in transcription, as "08" seems to be a typographical error for "320" (since $8 \times 40 = 320$ ).

STEP 2

First, we need to rewrite the equation in standard form, which means all terms should be on one side of the equation and set equal to zero.
$x^{3} + 8x^{2} - 40x - 320 = 0$

STEP 3

Look for a common factor in all terms of the polynomial. If there is one, factor it out. In this case, there is no common factor other than 1.

STEP 4

We will attempt to factor by grouping, which involves grouping terms to create common factors. Group the first two terms together and the last two terms together.
$(x^{3} + 8x^{2}) - (40x + 320) = 0$

STEP 5

Factor out the common factor from each group.
$x^{2}(x + 8) - 40(x + 8) = 0$

STEP 6

Now we see that $x + 8$ is a common factor. Factor $x + 8$ out of the entire expression.
$(x + 8)(x^{2} - 40) = 0$

STEP 7

Now we need to factor $x^{2} - 40$ further, if possible. Recognize that $x^{2} - 40$ is a difference of squares.
$x^{2} - 40 = x^{2} - (6.32455532033676)^{2}$

STEP 8

Factor the difference of squares.
$x^{2} - (6.32455532033676)^{2} = (x - 6.32455532033676)(x + 6.32455532033676)$

STEP 9

Now we have factored the original polynomial completely.
$(x + 8)(x - 6.32455532033676)(x + 6.32455532033676) = 0$

STEP 10

Apply the zero-product property, which states that if a product of factors is zero, then at least one of the factors must be zero.
$x + 8 = 0 \quad \text{or} \quad x - 6.32455532033676 = 0 \quad \text{or} \quad x + 6.32455532033676 = 0$

STEP 11

Solve each equation for $x$ .
For $x + 8 = 0$ : $x = -8$
For $x - 6.32455532033676 = 0$ : $x = 6.32455532033676$
For $x + 6.32455532033676 = 0$ : $x = -6.32455532033676$

STEP 12

Combine the solutions to form the solution set.
The solution set is $\{-8, -6.32455532033676, 6.32455532033676\}$ .
However, since we are dealing with a polynomial with integer coefficients, we expect the solutions to be either integers or irrational numbers that can be expressed in a simplified radical form. The decimal $6.32455532033676$ is an approximation of $\sqrt{40}$ , so we should express the solutions in exact form.

STEP 13

Recognize that $6.32455532033676$ is an approximation of $\sqrt{40}$ , which simplifies to $2\sqrt{10}$ .

STEP 14

Rewrite the solution set using the exact values.
The solution set is $\{-8, -2\sqrt{10}, 2\sqrt{10}\}$ .

STEP 15

Choose the correct option based on the solution set.
The correct choice is A. The solution set is $\{-8, -2\sqrt{10}, 2\sqrt{10}\}$ .

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