Westonci.ca offers fast, accurate answers to your questions. Join our community and get the insights you need now. Our platform offers a seamless experience for finding reliable answers from a network of knowledgeable professionals. Join our Q&A platform to connect with experts dedicated to providing accurate answers to your questions in various fields.
Sagot :
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]
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]
Thank you for choosing our service. We're dedicated to providing the best answers for all your questions. Visit us again. Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. Your questions are important to us at Westonci.ca. Visit again for expert answers and reliable information.
