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

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 $Q$. Our goal is to determine the operation that results in the simplified expression.

What are Polynomials?

A polynomial is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication. Polynomials can be classified based on the degree of the highest power of the variable. For example, a polynomial of degree 4 has the highest power of the variable as 4.

Properties of Polynomials

Polynomials have several properties that make them useful in algebra and other areas of mathematics. Some of the key properties include:

• Addition and Subtraction: Polynomials can be added and subtracted by combining like terms.
• Multiplication: Polynomials can be multiplied using the distributive property.
• Division: Polynomials can be divided using long division or synthetic division.

Simplifying Polynomials

Simplifying polynomials involves combining like terms and eliminating any unnecessary terms. This can be done using various techniques, including factoring, combining like terms, and canceling out common factors.

Factoring Polynomials

Factoring polynomials involves expressing them as a product of simpler polynomials. This can be done using various techniques, including:

• Greatest Common Factor (GCF): Factoring out the GCF of all terms in the polynomial.
• Difference of Squares: Factoring expressions of the form $a^2 - b^2$.
• Sum and Difference: Factoring expressions of the form $a^2 + 2ab + b^2$ and $a^2 - 2ab + b^2$.

Combining Like Terms

Combining like terms involves adding or subtracting terms that have the same variable and exponent. This can be done using the following steps:

1. Identify like terms: Identify terms that have the same variable and exponent.
2. Combine coefficients: Combine the coefficients of like terms by adding or subtracting them.
3. Simplify: Simplify the resulting expression by combining like terms.

Canceling Out Common Factors

Canceling out common factors involves dividing both the numerator and denominator by a common factor. This can be done using the following steps:

1. Identify common factors: Identify common factors in the numerator and denominator.
2. Divide: Divide both the numerator and denominator by the common factor.
3. Simplify: Simplify the resulting expression by canceling out the common factor.

Simplifying the Given Polynomials

Now that we have discussed the properties and operations of polynomials, let's apply these concepts to the given polynomials $P$ and $Q$.

Polynomial P

The polynomial $P$ is given by:

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

To simplify this polynomial, we can start by factoring out the GCF of all terms.

Factoring Out the GCF

The GCF of all terms in the polynomial $P$ is 1. Therefore, we can factor out 1 as follows:

$$P = 1(x^4 + 3x^3 + 2x^2 - x + 2)$$

Combining Like Terms

Now that we have factored out the GCF, we can combine like terms in the polynomial $P$.

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

The like terms in the polynomial $P$ are:

• $x^4$
• $3x^3$
• $2x^2$
• $-x$
• $2$

We can combine the coefficients of like terms by adding or subtracting them.

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

Simplifying the Polynomial P

Now that we have combined like terms, we can simplify the polynomial $P$ by canceling out any unnecessary terms.

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

The polynomial $P$ is already simplified, so we can stop here.

Polynomial Q

The polynomial $Q$ is given by:

$$Q = \left(x^3 + 2x^2 + 3\right)\left(x^2 - 2\right)$$

To simplify this polynomial, we can start by multiplying the two binomials using the distributive property.

Multiplying Binomials

The distributive property states that for any real numbers $a$, $b$, and $c$, we have:

$$a(b + c) = ab + ac$$

We can apply this property to the polynomial $Q$ as follows:

$$Q = \left(x^3 + 2x^2 + 3\right)\left(x^2 - 2\right)$$

$$Q = x^3(x^2 - 2) + 2x^2(x^2 - 2) + 3(x^2 - 2)$$

Simplifying the Polynomial Q

Now that we have multiplied the two binomials, we can simplify the polynomial $Q$ by combining like terms.

$$Q = x^3(x^2 - 2) + 2x^2(x^2 - 2) + 3(x^2 - 2)$$

The like terms in the polynomial $Q$ are:

• $x^3(x^2 - 2)$
• $2x^2(x^2 - 2)$
• $3(x^2 - 2)$

We can combine the coefficients of like terms by adding or subtracting them.

$$Q = x^5 - 2x^3 + 2x^4 - 4x^2 + 3x^2 - 6$$

Simplifying the Polynomial Q

Now that we have combined like terms, we can simplify the polynomial $Q$ by canceling out any unnecessary terms.

$$Q = x^5 - 2x^3 + 2x^4 - 4x^2 + 3x^2 - 6$$

The polynomial $Q$ is already simplified, so we can stop here.

Conclusion

In this article, we have discussed the properties and operations of polynomials, focusing on the given polynomials $P$ and $Q$. We have applied various techniques, including factoring, combining like terms, and canceling out common factors, to simplify the polynomials. The resulting simplified expressions are:

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

$$Q = x^5 - 2x^3 + 2x^4 - 4x^2 + 3x^2 - 6$$

These simplified expressions can be used in various mathematical applications, including algebra, calculus, and engineering.

Final Answer

The final answer is: $\boxed{P = x^4 + 3x^3 + 2x^2 - x + 2, Q = x^5 - 2x^3 + 2x^4 - 4x^2 + 3x^2 - 6}$

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 $Q$. Our goal is to determine the operation that results in the simplified expression.

Q&A: Simplifying Polynomials

Q: What is a polynomial?

A: A polynomial is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication. Polynomials can be classified based on the degree of the highest power of the variable.

Q: What are the properties of polynomials?

A: Polynomials have several properties that make them useful in algebra and other areas of mathematics. Some of the key properties include:

• Addition and Subtraction: Polynomials can be added and subtracted by combining like terms.
• Multiplication: Polynomials can be multiplied using the distributive property.
• Division: Polynomials can be divided using long division or synthetic division.

Q: How do I simplify a polynomial?

A: To simplify a polynomial, you can use various techniques, including factoring, combining like terms, and canceling out common factors.

Q: What is factoring?

A: Factoring involves expressing a polynomial as a product of simpler polynomials. This can be done using various techniques, including:

• Greatest Common Factor (GCF): Factoring out the GCF of all terms in the polynomial.
• Difference of Squares: Factoring expressions of the form $a^2 - b^2$.
• Sum and Difference: Factoring expressions of the form $a^2 + 2ab + b^2$ and $a^2 - 2ab + b^2$.

Q: How do I combine like terms?

A: To combine like terms, you can follow these steps:

1. Identify like terms: Identify terms that have the same variable and exponent.
2. Combine coefficients: Combine the coefficients of like terms by adding or subtracting them.
3. Simplify: Simplify the resulting expression by combining like terms.

Q: How do I cancel out common factors?

A: To cancel out common factors, you can follow these steps:

1. Identify common factors: Identify common factors in the numerator and denominator.
2. Divide: Divide both the numerator and denominator by the common factor.
3. Simplify: Simplify the resulting expression by canceling out the common factor.

Q: Can you provide an example of simplifying a polynomial?

A: Let's consider the polynomial $P = x^4 + 3x^3 + 2x^2 - x + 2$. To simplify this polynomial, we can start by factoring out the GCF of all terms.

Q: How do I simplify the polynomial $P = x^4 + 3x^3 + 2x^2 - x + 2$?

A: To simplify the polynomial $P = x^4 + 3x^3 + 2x^2 - x + 2$, we can follow these steps:

1. Factor out the GCF: The GCF of all terms in the polynomial $P$ is 1. Therefore, we can factor out 1 as follows:

$$P = 1(x^4 + 3x^3 + 2x^2 - x + 2)$$

2. Combine like terms: Now that we have factored out the GCF, we can combine like terms in the polynomial $P$.

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

The like terms in the polynomial $P$ are:

• $x^4$
• $3x^3$
• $2x^2$
• $-x$
• $2$

We can combine the coefficients of like terms by adding or subtracting them.

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

3. Simplify the polynomial: Now that we have combined like terms, we can simplify the polynomial $P$ by canceling out any unnecessary terms.

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

The polynomial $P$ is already simplified, so we can stop here.

Q: Can you provide an example of simplifying a polynomial with multiple terms?

A: Let's consider the polynomial $Q = \left(x^3 + 2x^2 + 3\right)\left(x^2 - 2\right)$. To simplify this polynomial, we can start by multiplying the two binomials using the distributive property.

Q: How do I simplify the polynomial $Q = \left(x^3 + 2x^2 + 3\right)\left(x^2 - 2\right)$?

A: To simplify the polynomial $Q = \left(x^3 + 2x^2 + 3\right)\left(x^2 - 2\right)$, we can follow these steps:

1. Multiply the binomials: We can multiply the two binomials using the distributive property.

$$Q = \left(x^3 + 2x^2 + 3\right)\left(x^2 - 2\right)$$

$$Q = x^3(x^2 - 2) + 2x^2(x^2 - 2) + 3(x^2 - 2)$$

2. Combine like terms: Now that we have multiplied the two binomials, we can combine like terms in the polynomial $Q$.

$$Q = x^3(x^2 - 2) + 2x^2(x^2 - 2) + 3(x^2 - 2)$$

The like terms in the polynomial $Q$ are:

• $x^3(x^2 - 2)$
• $2x^2(x^2 - 2)$
• $3(x^2 - 2)$

We can combine the coefficients of like terms by adding or subtracting them.

$$Q = x^5 - 2x^3 + 2x^4 - 4x^2 + 3x^2 - 6$$

3. Simplify the polynomial: Now that we have combined like terms, we can simplify the polynomial $Q$ by canceling out any unnecessary terms.

$$Q = x^5 - 2x^3 + 2x^4 - 4x^2 + 3x^2 - 6$$

The polynomial $Q$ is already simplified, so we can stop here.

Conclusion

In this article, we have discussed the properties and operations of polynomials, focusing on the given polynomials $P$ and $Q$. We have applied various techniques, including factoring, combining like terms, and canceling out common factors, to simplify the polynomials. The resulting simplified expressions are:

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

$$Q = x^5 - 2x^3 + 2x^4 - 4x^2 + 3x^2 - 6$$

These simplified expressions can be used in various mathematical applications, including algebra, calculus, and engineering.

Final Answer

The final answer is: $\boxed{P = x^4 + 3x^3 + 2x^2 - x + 2, Q = x^5 - 2x^3 + 2x^4 - 4x^2 + 3x^2 - 6}$