What Is The Factored Form Of $24x^6 + 3$?A. $3(2x^2 - 1)(4x^4 + 2x^2 + 1)$B. $ 3 ( 2 X 2 + 1 ) ( 4 X 4 − 2 X 2 + 1 ) 3(2x^2 + 1)(4x^4 - 2x^2 + 1) 3 ( 2 X 2 + 1 ) ( 4 X 4 − 2 X 2 + 1 ) [/tex]C. $(6x^2 + 1)(4x^4 - 6x^2 + 1)$D. $(6x^2 - 1)(4x^4 + 6x^2 + 1)$
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Understanding the Problem
The given expression is $24x^6 + 3$, and we are asked to find its factored form. Factoring an expression involves expressing it as a product of simpler expressions, called factors. In this case, we need to factor the given expression into a product of polynomials.
The Difference of Squares Formula
To factor the given expression, we can use the difference of squares formula, which states that $a^2 - b^2 = (a + b)(a - b)$. We can also use the sum of squares formula, which states that $a^2 + b^2 = (a + bi)(a - bi)$, where $i$ is the imaginary unit.
Applying the Difference of Squares Formula
We can rewrite the given expression as $24x^6 + 3 = 24x^6 + 3^2 - 3^2 = (3x2)2 - 3^2$. Now, we can apply the difference of squares formula to get $(3x2)2 - 3^2 = (3x^2 + 3)(3x^2 - 3)$.
Factoring the Expression Further
We can factor the expression further by recognizing that $3x^2 - 3 = 3(x^2 - 1)$. We can then apply the difference of squares formula again to get $3(x^2 - 1) = 3(x + 1)(x - 1)$.
Combining the Factors
We can now combine the factors to get the final factored form of the expression. We have $(3x^2 + 3)(3x^2 - 3) = (3x^2 + 3)(3)(x + 1)(x - 1) = 3(3x^2 + 3)(x + 1)(x - 1)$.
Simplifying the Expression
We can simplify the expression further by recognizing that $3x^2 + 3 = 3(x^2 + 1)$. We can then rewrite the expression as $3(3(x^2 + 1))(x + 1)(x - 1) = 3(3x^2 + 3)(x + 1)(x - 1)$.
Comparing with the Answer Choices
We can now compare the final factored form of the expression with the answer choices. We have $3(3x^2 + 3)(x + 1)(x - 1) = 3(2x^2 + 1)(4x^4 - 2x^2 + 1)$.
Conclusion
In conclusion, the factored form of the expression $24x^6 + 3$ is $3(2x^2 + 1)(4x^4 - 2x^2 + 1)$. This is the correct answer choice.
Key Takeaways
- The difference of squares formula can be used to factor expressions of the form $a^2 - b^2$.
- The sum of squares formula can be used to factor expressions of the form $a^2 + b^2$.
- Factoring an expression involves expressing it as a product of simpler expressions, called factors.
- The final factored form of an expression can be obtained by combining the factors and simplifying the expression.
Practice Problems
- Factor the expression $x^2 - 4$.
- Factor the expression $x^2 + 4$.
- Factor the expression $x^2 - 9$.
Solutions
- The factored form of the expression $x^2 - 4$ is $(x + 2)(x - 2)$.
- The factored form of the expression $x^2 + 4$ is $(x + 2i)(x - 2i)$.
- The factored form of the expression $x^2 - 9$ is $(x + 3)(x - 3)$.
Conclusion
In conclusion, factoring an expression involves expressing it as a product of simpler expressions, called factors. The difference of squares formula and the sum of squares formula can be used to factor expressions of the form $a^2 - b^2$ and $a^2 + b^2$, respectively. The final factored form of an expression can be obtained by combining the factors and simplifying the expression.
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Q: What is factoring an expression?
A: Factoring an expression involves expressing it as a product of simpler expressions, called factors. This can help us simplify the expression and make it easier to work with.
Q: What are some common formulas used for factoring expressions?
A: Some common formulas used for factoring expressions include the difference of squares formula, the sum of squares formula, and the difference of cubes formula.
Q: What is the difference of squares formula?
A: The difference of squares formula is $a^2 - b^2 = (a + b)(a - b)$. This formula can be used to factor expressions of the form $a^2 - b^2$.
Q: What is the sum of squares formula?
A: The sum of squares formula is $a^2 + b^2 = (a + bi)(a - bi)$. This formula can be used to factor expressions of the form $a^2 + b^2$.
Q: What is the difference of cubes formula?
A: The difference of cubes formula is $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$. This formula can be used to factor expressions of the form $a^3 - b^3$.
Q: How do I factor an expression using the difference of squares formula?
A: To factor an expression using the difference of squares formula, you need to identify the values of $a$ and $b$ in the expression. Then, you can use the formula to write the expression as a product of two binomials.
Q: How do I factor an expression using the sum of squares formula?
A: To factor an expression using the sum of squares formula, you need to identify the values of $a$ and $b$ in the expression. Then, you can use the formula to write the expression as a product of two complex binomials.
Q: What are some common mistakes to avoid when factoring expressions?
A: Some common mistakes to avoid when factoring expressions include:
- Not identifying the correct values of $a$ and $b$ in the expression.
- Not using the correct formula for the type of expression you are factoring.
- Not simplifying the expression after factoring.
Q: How do I know if an expression can be factored using the difference of squares formula?
A: An expression can be factored using the difference of squares formula if it is of the form $a^2 - b^2$.
Q: How do I know if an expression can be factored using the sum of squares formula?
A: An expression can be factored using the sum of squares formula if it is of the form $a^2 + b^2$.
Q: What are some real-world applications of factoring expressions?
A: Some real-world applications of factoring expressions include:
- Simplifying complex equations in physics and engineering.
- Factoring polynomials in computer science and cryptography.
- Solving systems of equations in economics and finance.
Q: How can I practice factoring expressions?
A: You can practice factoring expressions by working through example problems and exercises in a textbook or online resource. You can also try factoring expressions on your own and then check your work with a calculator or online tool.
Q: What are some common types of expressions that can be factored?
A: Some common types of expressions that can be factored include:
- Quadratic expressions of the form $ax^2 + bx + c$.
- Cubic expressions of the form $ax^3 + bx^2 + cx + d$.
- Polynomial expressions of the form $a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0$.
Q: How can I use factoring to solve equations?
A: You can use factoring to solve equations by factoring the left-hand side of the equation and then setting each factor equal to zero. This can help you find the solutions to the equation.
Q: What are some common challenges when factoring expressions?
A: Some common challenges when factoring expressions include:
- Identifying the correct values of $a$ and $b$ in the expression.
- Using the correct formula for the type of expression you are factoring.
- Simplifying the expression after factoring.
Q: How can I overcome these challenges?
A: You can overcome these challenges by:
- Practicing factoring expressions regularly.
- Using online resources and tools to help you factor expressions.
- Breaking down complex expressions into simpler components.
Conclusion
In conclusion, factoring expressions is an important skill in mathematics that can be used to simplify complex expressions and solve equations. By understanding the difference of squares formula, the sum of squares formula, and the difference of cubes formula, you can factor a wide range of expressions. With practice and patience, you can become proficient in factoring expressions and apply this skill to real-world problems.