Select The Correct Answer.Consider The Polynomials Given Below:$\[ \begin{align*} P &= X^4 + 3x^3 + 2x^2 - X + 2 \\ Q &= (x^8 + 2x^2 + 3)(x^2 - 2) \end{align*} \\]Determine The Operation That Results In The Simplified Expression Below:$\[

Understanding the Problem

When dealing with polynomials, it's essential to understand the properties and operations that can be performed on them. In this article, we will explore the process of simplifying polynomials, focusing on the given polynomials $P$ and $Q$. We will determine the operation that results in the simplified expression.

Given Polynomials

The given polynomials are:

{ \begin{align*} P &= x^4 + 3x^3 + 2x^2 - x + 2 \\ Q &= (x^8 + 2x^2 + 3)(x^2 - 2) \end{align*} \}

Simplifying Polynomial Q

To simplify polynomial $Q$, we need to expand the expression using the distributive property. This involves multiplying each term in the first polynomial by each term in the second polynomial.

{ \begin{align*} Q &= (x^8 + 2x^2 + 3)(x^2 - 2) \\ &= x^8(x^2 - 2) + 2x^2(x^2 - 2) + 3(x^2 - 2) \\ &= x^{10} - 2x^8 + 2x^4 - 4x^2 + 3x^2 - 6 \\ &= x^{10} - 2x^8 + 2x^4 - x^2 - 6 \end{align*} \}

Comparing Polynomials P and Q

Now that we have simplified polynomial $Q$, we can compare it with polynomial $P$. We need to determine the operation that results in the simplified expression.

{ \begin{align*} P &= x^4 + 3x^3 + 2x^2 - x + 2 \\ Q &= x^{10} - 2x^8 + 2x^4 - x^2 - 6 \end{align*} \}

Determining the Operation

To determine the operation that results in the simplified expression, we need to examine the terms of both polynomials. We can see that polynomial $Q$ has a term $x^{10}$, which is not present in polynomial $P$. This suggests that the operation involves adding or subtracting polynomials.

Subtracting Polynomial P from Q

One possible operation is to subtract polynomial $P$ from polynomial $Q$. This involves subtracting each term of polynomial $P$ from the corresponding term of polynomial $Q$.

{ \begin{align*} Q - P &= (x^{10} - 2x^8 + 2x^4 - x^2 - 6) - (x^4 + 3x^3 + 2x^2 - x + 2) \\ &= x^{10} - 2x^8 + 2x^4 - x^2 - 6 - x^4 - 3x^3 - 2x^2 + x + 2 \\ &= x^{10} - 2x^8 - 3x^3 + x^4 - 3x^2 + x - 4 \end{align*} \}

Conclusion

In conclusion, the operation that results in the simplified expression is subtracting polynomial $P$ from polynomial $Q$. This involves subtracting each term of polynomial $P$ from the corresponding term of polynomial $Q$. The resulting expression is $x^{10} - 2x^8 - 3x^3 + x^4 - 3x^2 + x - 4$.

Final Answer

The final answer is: $\boxed{x^{10} - 2x^8 - 3x^3 + x^4 - 3x^2 + x - 4}$

Understanding the Basics

Polynomials are algebraic expressions consisting of variables and coefficients combined using only addition, subtraction, and multiplication. Simplifying polynomials involves combining like terms and performing operations to obtain a simpler expression.

Q: What is the difference between a polynomial and an expression?

A: A polynomial is a specific type of algebraic expression that consists of variables and coefficients combined using only addition, subtraction, and multiplication. An expression, on the other hand, can be any combination of variables, coefficients, and mathematical operations.

Q: How do I simplify a polynomial?

A: To simplify a polynomial, you need to combine like terms and perform operations to obtain a simpler expression. This involves adding or subtracting terms with the same variable and exponent.

Q: What are like terms?

A: Like terms are terms that have the same variable and exponent. For example, $2x^2$ and $3x^2$ are like terms because they both have the variable $x$ and the exponent $2$.

Q: How do I combine like terms?

A: To combine like terms, you need to add or subtract the coefficients of the terms. For example, if you have the terms $2x^2$ and $3x^2$, you can combine them by adding the coefficients: $2x^2 + 3x^2 = 5x^2$.

Q: What is the distributive property?

A: The distributive property is a mathematical property that allows you to multiply a single term by multiple terms. For example, if you have the expression $2(x + 3)$, you can use the distributive property to multiply the term $2$ by each term inside the parentheses: $2(x + 3) = 2x + 6$.

Q: How do I expand a polynomial?

A: To expand a polynomial, you need to use the distributive property to multiply each term by each other term. For example, if you have the polynomial $(x + 2)(x + 3)$, you can expand it by multiplying each term by each other term: $(x + 2)(x + 3) = x^2 + 3x + 2x + 6 = x^2 + 5x + 6$.

Q: What is the difference between a polynomial and a rational expression?

A: A polynomial is an algebraic expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication. A rational expression, on the other hand, is an algebraic expression that involves division by a variable or a constant.

Q: How do I simplify a rational expression?

A: To simplify a rational expression, you need to combine like terms and perform operations to obtain a simpler expression. This involves adding or subtracting terms with the same variable and exponent, and also canceling out common factors in the numerator and denominator.

Q: What is the final answer to the original problem?

A: The final answer to the original problem is $x^{10} - 2x^8 - 3x^3 + x^4 - 3x^2 + x - 4$.

Conclusion

In conclusion, simplifying polynomials involves combining like terms and performing operations to obtain a simpler expression. By understanding the basics of polynomials and using the distributive property, you can simplify complex expressions and obtain a final answer.

Final Answer

The final answer is: $\boxed{x^{10} - 2x^8 - 3x^3 + x^4 - 3x^2 + x - 4}$