To solve the polynomial equation [tex]\( x^4 - 81 = 0 \)[/tex], let's go through the steps methodically.
1. Rewrite the Equation:
The given polynomial equation is:
[tex]\[
x^4 - 81 = 0
\][/tex]
2. Recognize It as a Difference of Squares:
Notice that [tex]\( 81 \)[/tex] can be written as [tex]\( 9^2 \)[/tex]. Also, recall that every difference of squares can be factored as:
[tex]\[
a^2 - b^2 = (a - b)(a + b)
\][/tex]
Here, [tex]\( x^4 \)[/tex] is [tex]\( (x^2)^2 \)[/tex], and [tex]\( 81 \)[/tex] is [tex]\( 9^2 \)[/tex]. Applying the difference of squares:
[tex]\[
x^4 - 81 = (x^2)^2 - 9^2 = (x^2 - 9)(x^2 + 9)
\][/tex]
3. Factor Further:
Now, we need to factor each part further if possible.
- For [tex]\( x^2 - 9 \)[/tex]:
[tex]\[
x^2 - 9 = (x - 3)(x + 3)
\][/tex]
- For [tex]\( x^2 + 9 \)[/tex]:
It can't be factored further using real numbers, but it can be expressed using imaginary numbers:
[tex]\[
x^2 + 9 = (x - 3i)(x + 3i)
\][/tex]
where [tex]\( i \)[/tex] is the imaginary unit ([tex]\( i^2 = -1 \)[/tex]).
4. Combine All Factors:
Putting it all together, we have:
[tex]\[
x^4 - 81 = (x - 3)(x + 3)(x - 3i)(x + 3i)
\][/tex]
5. Find the Roots:
Setting each factor equal to zero gives the solutions to the equation:
[tex]\[
x - 3 = 0 \quad \Rightarrow \quad x = 3
\][/tex]
[tex]\[
x + 3 = 0 \quad \Rightarrow \quad x = -3
\][/tex]
[tex]\[
x - 3i = 0 \quad \Rightarrow \quad x = 3i
\][/tex]
[tex]\[
x + 3i = 0 \quad \Rightarrow \quad x = -3i
\][/tex]
6. List the Solutions:
The complete set of solutions to the polynomial [tex]\( x^4 - 81 = 0 \)[/tex] is:
[tex]\[
x = 3, -3, 3i, -3i
\][/tex]
By examining the options provided:
a. [tex]\( -3, 0, 3 \)[/tex] – This is incorrect because it includes [tex]\( 0 \)[/tex], which is not a solution, and misses the imaginary solutions.
b. [tex]\( 3, -3, 3i, -3i \)[/tex] – This is the correct option as it includes all and only the solutions.
c. [tex]\( 81 \)[/tex] – This is incorrect because it represents a value, not a solution to the polynomial in question.
d. [tex]\( -9, 9 \)[/tex] – This is incorrect because these values do not solve the polynomial [tex]\( x^4 - 81 = 0 \)[/tex].
Therefore, the correct answer is:
b. [tex]\( 3, -3, 3i, -3i \)[/tex]
