Simplify The Following Expression:${ x 2(x 2 + X + 1) }$ { x^{[3]} + X + X \}

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Introduction

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In this article, we will simplify the given mathematical expression, which involves algebraic manipulation and understanding of mathematical operations. The expression to be simplified is $x^2(x^2 + x + 1)$ and $x^{[3]} + x + x$. We will break down the expression step by step and simplify it to its simplest form.

Understanding the Expression

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The given expression consists of two parts: $x^2(x^2 + x + 1)$ and $x^{[3]} + x + x$. The first part is a quadratic expression multiplied by $x^2$, while the second part involves exponentiation and addition.

Quadratic Expression

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The quadratic expression $x^2 + x + 1$ is a polynomial of degree 2. It can be factored or simplified using various algebraic techniques.

Exponentiation

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The expression $x^{[3]}$ represents the cube of $x$, which is equivalent to $x^3$. This is a common notation used in mathematics to represent exponentiation.

Simplifying the Expression

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To simplify the given expression, we will start by expanding the quadratic expression and then combine like terms.

Expanding the Quadratic Expression

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We can expand the quadratic expression $x^2 + x + 1$ by multiplying each term by $x^2$.

$$x^2(x^2 + x + 1) = x^2 \cdot x^2 + x^2 \cdot x + x^2 \cdot 1$$

Using the properties of exponents, we can simplify the expression further.

$$x^2 \cdot x^2 = x^4$$

$$x^2 \cdot x = x^3$$

$$x^2 \cdot 1 = x^2$$

Therefore, the expanded expression becomes:

$$x^4 + x^3 + x^2$$

Combining Like Terms

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Now that we have expanded the quadratic expression, we can combine like terms to simplify the expression further.

$$x^4 + x^3 + x^2 = x^4 + x^3 + x^2$$

There are no like terms to combine, so the expression remains the same.

Simplifying the Exponentiation

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The expression $x^{[3]} + x + x$ can be simplified by evaluating the exponentiation.

$$x^{[3]} = x^3$$

Therefore, the expression becomes:

$$x^3 + x + x$$

Using the properties of addition, we can combine like terms.

$$x^3 + x + x = x^3 + 2x$$

Conclusion

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In this article, we simplified the given mathematical expression by expanding the quadratic expression and combining like terms. We also evaluated the exponentiation and combined like terms to simplify the expression further. The final simplified expression is $x^4 + x^3 + x^2$ and $x^3 + 2x$.

Final Answer

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The final answer is:

• $x^4 + x^3 + x^2$
• $x^3 + 2x$

Note: The final answer is in two parts, as the original expression consisted of two separate parts.

Discussion

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The given expression involves algebraic manipulation and understanding of mathematical operations. The expression can be simplified using various algebraic techniques, such as expanding and combining like terms. The final simplified expression provides a clear understanding of the original expression and can be used as a reference for future mathematical calculations.

Related Topics

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• Algebraic manipulation
• Exponentiation
• Quadratic expressions
• Like terms

References

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• [1] Algebraic manipulation, Wikipedia.
• [2] Exponentiation, Wikipedia.
• [3] Quadratic expressions, Wikipedia.
• [4] Like terms, Wikipedia.

Note: The references provided are for general information and are not specific to the given expression.

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Introduction

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In the previous article, we simplified the given mathematical expression by expanding the quadratic expression and combining like terms. In this article, we will address some frequently asked questions (FAQs) related to simplifying mathematical expressions.

Q&A

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Q1: What is the difference between expanding and simplifying a mathematical expression?

A1: Expanding a mathematical expression involves breaking it down into its individual components, while simplifying an expression involves combining like terms and eliminating any unnecessary components.

Q2: How do I know when to expand or simplify a mathematical expression?

A2: You should expand a mathematical expression when you need to break it down into its individual components, such as when you are working with a complex expression or when you need to identify specific terms. You should simplify an expression when you need to combine like terms and eliminate any unnecessary components, such as when you are working with a large expression or when you need to make the expression more manageable.

Q3: What is the difference between a quadratic expression and a polynomial expression?

A3: A quadratic expression is a polynomial expression of degree 2, which means it has a highest power of 2. A polynomial expression, on the other hand, can have any degree, including 2.

Q4: How do I simplify a quadratic expression?

A4: To simplify a quadratic expression, you can use various algebraic techniques, such as factoring, completing the square, or using the quadratic formula.

Q5: What is the difference between an exponent and a power?

A5: An exponent is a small number that is raised to a power, while a power is the result of raising a number to a certain exponent. For example, in the expression $x^2$, the 2 is the exponent and the $x$ is the base.

Q6: How do I simplify an expression with exponents?

A6: To simplify an expression with exponents, you can use the properties of exponents, such as the product rule, the power rule, or the quotient rule.

Q7: What is the difference between a like term and an unlike term?

A7: A like term is a term that has the same variable and exponent, while an unlike term is a term that has a different variable or exponent.

Q8: How do I combine like terms?

A8: To combine like terms, you can add or subtract the coefficients of the like terms, while keeping the variable and exponent the same.

Q9: What is the final answer to the given expression?

A9: The final answer to the given expression is $x^4 + x^3 + x^2$ and $x^3 + 2x$.

Q10: Can you provide more examples of simplifying mathematical expressions?

A10: Yes, here are a few more examples:

• Simplify the expression $x^2 + 3x + 2$.
• Simplify the expression $x^3 - 2x^2 + x$.
• Simplify the expression $x^4 + 2x^3 - 3x^2$.

Conclusion

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In this article, we addressed some frequently asked questions (FAQs) related to simplifying mathematical expressions. We provided explanations and examples to help clarify the concepts and techniques involved in simplifying expressions.

Final Answer

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The final answer to the given expression is $x^4 + x^3 + x^2$ and $x^3 + 2x$.

Related Topics

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• Algebraic manipulation
• Exponentiation
• Quadratic expressions
• Like terms

References

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• [1] Algebraic manipulation, Wikipedia.
• [2] Exponentiation, Wikipedia.
• [3] Quadratic expressions, Wikipedia.
• [4] Like terms, Wikipedia.

Note: The references provided are for general information and are not specific to the given expression.