Which Of The Following Is Equivalent To $x^4-1$?A. $(x-1)^4$B. $\left(x^2-1\right)^2$C. \$(x-1)^2(x+1)^2$[/tex\]D. $(x-1)(x+1)\left(x^2+1\right)$

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Understanding the Problem

The problem requires us to find an equivalent expression for the given polynomial x4−1x^4-1. This means we need to find an expression that has the same value as x4−1x^4-1 for all values of xx. To solve this problem, we can use various algebraic techniques such as factoring, expanding, and simplifying expressions.

Option A: (x−1)4(x-1)^4

Let's start by analyzing option A: (x−1)4(x-1)^4. This expression can be expanded using the binomial theorem, which states that for any positive integer nn, (a+b)n=∑k=0n(nk)an−kbk(a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k. Applying this theorem to our expression, we get:

(x−1)4=x4−4x3+6x2−4x+1(x-1)^4 = x^4 - 4x^3 + 6x^2 - 4x + 1

However, this expression is not equivalent to x4−1x^4-1 because it has additional terms that are not present in the original expression.

Option B: (x2−1)2\left(x^2-1\right)^2

Next, let's analyze option B: (x2−1)2\left(x^2-1\right)^2. This expression can be expanded using the formula (a−b)2=a2−2ab+b2(a-b)^2 = a^2 - 2ab + b^2. Applying this formula to our expression, we get:

(x2−1)2=x4−2x2+1\left(x^2-1\right)^2 = x^4 - 2x^2 + 1

However, this expression is not equivalent to x4−1x^4-1 because it has an additional term −2x2-2x^2 that is not present in the original expression.

**Option C: $(x-1)2(x+1)2$

Now, let's analyze option C: $(x-1)2(x+1)2$. This expression can be expanded using the formula (a−b)(a+b)=a2−b2(a-b)(a+b) = a^2 - b^2. Applying this formula to our expression, we get:

(x−1)2(x+1)2=(x2−1)2=x4−2x2+1(x-1)^2(x+1)^2 = (x^2 - 1)^2 = x^4 - 2x^2 + 1

However, this expression is not equivalent to x4−1x^4-1 because it has an additional term −2x2-2x^2 that is not present in the original expression.

**Option D: $(x-1)(x+1)\left(x^2+1\right)$

Finally, let's analyze option D: $(x-1)(x+1)\left(x^2+1\right)$. This expression can be expanded using the formula (a−b)(a+b)=a2−b2(a-b)(a+b) = a^2 - b^2. Applying this formula to our expression, we get:

(x−1)(x+1)(x2+1)=(x2−1)(x2+1)=x4−1(x-1)(x+1)\left(x^2+1\right) = (x^2 - 1)\left(x^2+1\right) = x^4 - 1

This expression is equivalent to x4−1x^4-1 because it has the same value for all values of xx.

Conclusion

In conclusion, the correct answer is option D: $(x-1)(x+1)\left(x^2+1\right)$. This expression is equivalent to x4−1x^4-1 because it has the same value for all values of xx. The other options are not equivalent to x4−1x^4-1 because they have additional terms that are not present in the original expression.

Why is this important?

Understanding the concept of equivalent expressions is important in mathematics because it allows us to simplify complex expressions and solve problems more easily. In this case, we were able to find an equivalent expression for x4−1x^4-1 by using various algebraic techniques. This knowledge can be applied to a wide range of mathematical problems and is an essential tool for any mathematician or scientist.

Real-world applications

The concept of equivalent expressions has many real-world applications. For example, in physics, equivalent expressions can be used to simplify complex equations and solve problems related to motion and energy. In engineering, equivalent expressions can be used to design and optimize systems, such as electrical circuits and mechanical systems. In finance, equivalent expressions can be used to calculate interest rates and investment returns.

Final thoughts

In conclusion, the correct answer is option D: $(x-1)(x+1)\left(x^2+1\right)$. This expression is equivalent to x4−1x^4-1 because it has the same value for all values of xx. The other options are not equivalent to x4−1x^4-1 because they have additional terms that are not present in the original expression. Understanding the concept of equivalent expressions is important in mathematics because it allows us to simplify complex expressions and solve problems more easily.

Q: What is an equivalent expression?

A: An equivalent expression is an expression that has the same value as another expression for all values of the variables involved. In other words, two expressions are equivalent if they have the same output for every possible input.

Q: Why are equivalent expressions important?

A: Equivalent expressions are important because they allow us to simplify complex expressions and solve problems more easily. By finding an equivalent expression, we can often make the problem more manageable and easier to solve.

Q: How do I find an equivalent expression?

A: There are several ways to find an equivalent expression, including:

  • Factoring: This involves breaking down an expression into simpler factors.
  • Expanding: This involves multiplying out an expression to get a simpler form.
  • Simplifying: This involves combining like terms and eliminating any unnecessary terms.
  • Using algebraic identities: This involves using known formulas to simplify an expression.

Q: What are some common algebraic identities?

A: Some common algebraic identities include:

  • (a+b)2=a2+2ab+b2(a+b)^2 = a^2 + 2ab + b^2
  • (a−b)2=a2−2ab+b2(a-b)^2 = a^2 - 2ab + b^2
  • (a+b)(a−b)=a2−b2(a+b)(a-b) = a^2 - b^2
  • (a−b)(a+b)=a2−b2(a-b)(a+b) = a^2 - b^2

Q: How do I know if two expressions are equivalent?

A: To determine if two expressions are equivalent, you can try the following:

  • Plug in some values for the variables and see if the expressions produce the same output.
  • Use algebraic manipulations to simplify both expressions and see if they become identical.
  • Use a calculator or computer program to evaluate both expressions and see if they produce the same result.

Q: Can equivalent expressions be used in real-world applications?

A: Yes, equivalent expressions can be used in a wide range of real-world applications, including:

  • Physics: Equivalent expressions can be used to simplify complex equations and solve problems related to motion and energy.
  • Engineering: Equivalent expressions can be used to design and optimize systems, such as electrical circuits and mechanical systems.
  • Finance: Equivalent expressions can be used to calculate interest rates and investment returns.

Q: Are equivalent expressions only used in mathematics?

A: No, equivalent expressions are not only used in mathematics. They can be used in a wide range of fields, including science, engineering, economics, and finance.

Q: Can equivalent expressions be used to solve complex problems?

A: Yes, equivalent expressions can be used to solve complex problems. By finding an equivalent expression, you can often make the problem more manageable and easier to solve.

Q: How do I get started with using equivalent expressions?

A: To get started with using equivalent expressions, try the following:

  • Start with simple expressions and try to find equivalent expressions by factoring, expanding, or simplifying.
  • Practice using algebraic identities to simplify expressions.
  • Try plugging in some values for the variables and see if the expressions produce the same output.
  • Use a calculator or computer program to evaluate both expressions and see if they produce the same result.

Q: Are there any resources available to help me learn more about equivalent expressions?

A: Yes, there are many resources available to help you learn more about equivalent expressions, including:

  • Textbooks and online resources that provide examples and exercises.
  • Online tutorials and videos that demonstrate how to use equivalent expressions.
  • Practice problems and quizzes that can help you test your skills.
  • Online communities and forums where you can ask questions and get help from others.