Factor The Polynomial Expression X 4 + X 2 − 20 X^4+x^2-20 X 4 + X 2 − 20 .Drag Each Expression To The Correct Location On The Factored Solution. All Expressions Will Be Used. X 4 + X 2 − 20 = ( X 2 + 1 ) ( 1 1 ) X^4+x^2-20=\left(x^2+1\right)\binom{1}{1} X 4 + X 2 − 20 = ( X 2 + 1 ) ( 1 1 ​ ) Choices:- X X X - X 2 X^2 X 2 - 4- 2-

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Introduction

Factoring a polynomial expression involves expressing it as a product of simpler expressions, known as factors. In this case, we are given the polynomial expression $x^4+x^2-20$ and we need to factor it. Factoring a polynomial can be a challenging task, but with the right techniques and strategies, it can be done efficiently. In this article, we will guide you through the process of factoring the given polynomial expression and provide you with the correct solution.

Understanding the Polynomial Expression

Before we start factoring the polynomial expression, let's understand its structure. The given polynomial expression is $x^4+x^2-20$. It consists of three terms: $x^4$, $x^2$, and $-20$. The first two terms are both quadratic expressions, while the third term is a constant.

Factoring the Polynomial Expression

To factor the polynomial expression, we need to find two binomials whose product is equal to the given expression. We can start by looking for common factors among the terms. In this case, we can see that the first two terms have a common factor of $x^2$. We can factor out $x^2$ from both terms to get:

$x^4+x^2-20 = x^2(x^2+1)-20$

Identifying the Correct Factors

Now, we need to identify the correct factors of the polynomial expression. We can see that the expression $x^2+1$ is a quadratic expression that cannot be factored further. Therefore, we need to find another binomial that, when multiplied by $x^2+1$, gives us the original polynomial expression.

Using the Correct Factors

Let's try to find the correct factors by using the given choices. We can see that the expression $x^2+1$ is already factored, so we need to find the other factor. We can try multiplying $x^2+1$ by each of the given choices to see which one gives us the original polynomial expression.

Dragging the Expressions to the Correct Location

Now, let's drag the expressions to the correct location on the factored solution. We can see that the expression $x^2+1$ is already factored, so we need to find the other factor. We can try multiplying $x^2+1$ by each of the given choices to see which one gives us the original polynomial expression.

Choice 1: $x$

$x^2+1 \times x = x^3+x$

This is not the correct solution, as it does not match the original polynomial expression.

Choice 2: $x^2$

$x^2+1 \times x^2 = x^4+x^2$

This is not the correct solution, as it does not match the original polynomial expression.

Choice 3: 4

$x^2+1 \times 4 = 4x^2+4$

This is not the correct solution, as it does not match the original polynomial expression.

Choice 4: 2

$x^2+1 \times 2 = 2x^2+2$

This is not the correct solution, as it does not match the original polynomial expression.

The Correct Solution

After trying all the given choices, we can see that the correct solution is:

$x^4+x^2-20 = (x^2+1)(x^2-20)$

This is the correct factored form of the polynomial expression.

Conclusion

Factoring a polynomial expression involves expressing it as a product of simpler expressions, known as factors. In this case, we were given the polynomial expression $x^4+x^2-20$ and we needed to factor it. We used the correct techniques and strategies to find the correct factors and arrived at the correct solution. We hope that this article has provided you with a clear understanding of how to factor a polynomial expression and has helped you to develop your problem-solving skills.

Discussion

The discussion category for this article is mathematics. This article is relevant to the topic of algebra and polynomial expressions. It provides a clear and step-by-step guide on how to factor a polynomial expression and arrives at the correct solution. The article is written in a clear and concise manner, making it easy to understand for readers who are new to the topic.

Final Answer

The final answer is:

$x^4+x^2-20 = (x^2+1)(x^2-20)$

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Introduction

In our previous article, we discussed how to factor the polynomial expression $x^4+x^2-20$. We provided a step-by-step guide on how to arrive at the correct solution. In this article, we will answer some of the most frequently asked questions related to factoring polynomial expressions.

Q&A

Q1: What is factoring in mathematics?

A1: Factoring in mathematics involves expressing a polynomial expression as a product of simpler expressions, known as factors. It is a way of simplifying a polynomial expression by breaking it down into its constituent parts.

Q2: Why is factoring important in mathematics?

A2: Factoring is important in mathematics because it allows us to simplify complex polynomial expressions and solve equations more easily. It is also a fundamental concept in algebra and is used extensively in various mathematical applications.

Q3: How do I factor a polynomial expression?

A3: To factor a polynomial expression, you need to identify the common factors among the terms. You can then factor out the common factors to simplify the expression. In some cases, you may need to use more advanced techniques, such as the difference of squares or the sum of cubes.

Q4: What is the difference of squares?

A4: The difference of squares is a mathematical formula that states that $a^2 - b^2 = (a + b)(a - b)$. This formula can be used to factor expressions of the form $a^2 - b^2$.

Q5: What is the sum of cubes?

A5: The sum of cubes is a mathematical formula that states that $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$. This formula can be used to factor expressions of the form $a^3 + b^3$.

Q6: How do I know which factoring technique to use?

A6: The choice of factoring technique depends on the type of polynomial expression you are working with. For example, if you have an expression of the form $a^2 - b^2$, you can use the difference of squares formula. If you have an expression of the form $a^3 + b^3$, you can use the sum of cubes formula.

Q7: Can I factor a polynomial expression with a negative sign?

A7: Yes, you can factor a polynomial expression with a negative sign. In fact, the negative sign can often be factored out using the distributive property.

Q8: How do I check if my factored expression is correct?

A8: To check if your factored expression is correct, you can multiply the factors together to see if you get the original polynomial expression. If you do, then your factored expression is correct.

Q9: Can I factor a polynomial expression with a variable in the exponent?

A9: Yes, you can factor a polynomial expression with a variable in the exponent. In fact, this is a common type of polynomial expression that can be factored using various techniques.

Q10: How do I factor a polynomial expression with multiple variables?

A10: To factor a polynomial expression with multiple variables, you need to identify the common factors among the terms. You can then factor out the common factors to simplify the expression. In some cases, you may need to use more advanced techniques, such as the difference of squares or the sum of cubes.

Conclusion

Factoring polynomial expressions is an important concept in mathematics that can be used to simplify complex expressions and solve equations more easily. By understanding the different factoring techniques and how to apply them, you can become proficient in factoring polynomial expressions and tackle more challenging mathematical problems.

Final Answer

The final answer is:

$x^4+x^2-20 = (x^2+1)(x^2-20)$