Simplify The Expression:${ (8x^4 + 7x^3 + 6x^2) - (-5x^4 + 3x^3 - 2x^2) }$

Introduction

In this article, we will simplify the given expression using basic algebraic operations. The expression involves addition and subtraction of polynomials, which can be simplified by combining like terms. We will use the distributive property and combine like terms to simplify the expression.

Understanding the Expression

The given expression is:

{ (8x^4 + 7x^3 + 6x^2) - (-5x^4 + 3x^3 - 2x^2) \}

This expression consists of two polynomials, one inside the parentheses and the other outside. The first polynomial is $8x^4 + 7x^3 + 6x^2$, and the second polynomial is $-5x^4 + 3x^3 - 2x^2$. We need to simplify this expression by combining like terms.

Step 1: Distribute the Negative Sign

The first step is to distribute the negative sign to the terms inside the second polynomial. This will change the sign of each term in the second polynomial.

{ (8x^4 + 7x^3 + 6x^2) - (-5x^4 + 3x^3 - 2x^2) \}

$= (8x^4 + 7x^3 + 6x^2) + (5x^4 - 3x^3 + 2x^2)$

Step 2: Combine Like Terms

Now that we have distributed the negative sign, we can combine like terms. Like terms are terms that have the same variable and exponent. In this case, we have four like terms: $8x^4$, $5x^4$, $7x^3$, and $-3x^3$. We can combine these terms by adding or subtracting their coefficients.

{ (8x^4 + 7x^3 + 6x^2) + (5x^4 - 3x^3 + 2x^2) \}

$= (8x^4 + 5x^4) + (7x^3 - 3x^3) + (6x^2 + 2x^2)$

Step 3: Simplify the Expression

Now that we have combined like terms, we can simplify the expression by adding or subtracting the coefficients.

{ (8x^4 + 5x^4) + (7x^3 - 3x^3) + (6x^2 + 2x^2) \}

$= 13x^4 + 4x^3 + 8x^2$

Conclusion

In this article, we simplified the given expression using basic algebraic operations. We distributed the negative sign to the terms inside the second polynomial and then combined like terms. Finally, we simplified the expression by adding or subtracting the coefficients. The simplified expression is $13x^4 + 4x^3 + 8x^2$.

Key Takeaways

• Distribute the negative sign to the terms inside the second polynomial.
• Combine like terms by adding or subtracting their coefficients.
• Simplify the expression by adding or subtracting the coefficients.

Real-World Applications

Simplifying expressions is an essential skill in mathematics and has many real-world applications. For example, in physics, we use algebraic expressions to describe the motion of objects. In engineering, we use algebraic expressions to design and analyze systems. In economics, we use algebraic expressions to model and analyze economic systems.

Common Mistakes

When simplifying expressions, it's easy to make mistakes. Here are some common mistakes to avoid:

• Not distributing the negative sign to the terms inside the second polynomial.
• Not combining like terms.
• Not simplifying the expression by adding or subtracting the coefficients.

Tips and Tricks

Here are some tips and tricks to help you simplify expressions:

• Use the distributive property to distribute the negative sign to the terms inside the second polynomial.
• Use the commutative property to rearrange the terms in the expression.
• Use the associative property to group the terms in the expression.

Conclusion

Introduction

In our previous article, we simplified the given expression using basic algebraic operations. In this article, we will answer some frequently asked questions about simplifying expressions.

Q: What is the distributive property?

A: The distributive property is a fundamental concept in algebra that allows us to distribute a coefficient to the terms inside a polynomial. For example, if we have the expression $2(x + y)$, we can use the distributive property to distribute the coefficient 2 to the terms inside the parentheses: $2x + 2y$.

Q: How do I distribute the negative sign?

A: To distribute the negative sign, we need to change the sign of each term inside the polynomial. For example, if we have the expression $-2x + 3y$, we can distribute the negative sign to the terms inside the polynomial: $-2x - 3y$.

Q: What are like terms?

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

Q: How do I combine like terms?

A: To combine like terms, we need to add or subtract their coefficients. For example, if we have the expression $2x^2 + 3x^2$, we can combine the like terms by adding their coefficients: $5x^2$.

Q: What is the commutative property?

A: The commutative property is a fundamental concept in algebra that allows us to rearrange the terms in an expression without changing its value. For example, if we have the expression $2x + 3y$, we can use the commutative property to rearrange the terms: $3y + 2x$.

Q: What is the associative property?

A: The associative property is a fundamental concept in algebra that allows us to group the terms in an expression without changing its value. For example, if we have the expression $(2x + 3y) + 4z$, we can use the associative property to group the terms: $2x + (3y + 4z)$.

Q: How do I simplify an expression?

A: To simplify an expression, we need to use the distributive property, combine like terms, and use the commutative and associative properties to rearrange the terms. For example, if we have the expression $2x^2 + 3x^2 + 4x^2$, we can simplify it by combining the like terms: $9x^2$.

Q: What are some common mistakes to avoid when simplifying expressions?

A: Some common mistakes to avoid when simplifying expressions include:

• Not distributing the negative sign to the terms inside the second polynomial.
• Not combining like terms.
• Not simplifying the expression by adding or subtracting the coefficients.

Q: How can I practice simplifying expressions?

A: You can practice simplifying expressions by working through examples and exercises. You can also use online resources and tools to help you practice and improve your skills.

Conclusion

In conclusion, simplifying expressions is an essential skill in mathematics that has many real-world applications. By understanding the distributive property, combining like terms, and using the commutative and associative properties, we can simplify complex expressions and make them easier to work with. Remember to avoid common mistakes and practice regularly to improve your skills.

Key Takeaways

• Distribute the negative sign to the terms inside the second polynomial.
• Combine like terms by adding or subtracting their coefficients.
• Use the commutative and associative properties to rearrange the terms.
• Avoid common mistakes such as not distributing the negative sign, not combining like terms, and not simplifying the expression.

Real-World Applications

Simplifying expressions is an essential skill in mathematics that has many real-world applications. For example, in physics, we use algebraic expressions to describe the motion of objects. In engineering, we use algebraic expressions to design and analyze systems. In economics, we use algebraic expressions to model and analyze economic systems.

Common Mistakes

When simplifying expressions, it's easy to make mistakes. Here are some common mistakes to avoid:

• Not distributing the negative sign to the terms inside the second polynomial.
• Not combining like terms.
• Not simplifying the expression by adding or subtracting the coefficients.

Tips and Tricks

Here are some tips and tricks to help you simplify expressions:

• Use the distributive property to distribute the negative sign to the terms inside the second polynomial.
• Use the commutative property to rearrange the terms in the expression.
• Use the associative property to group the terms in the expression.
• Practice regularly to improve your skills.