Simplify The Expression:$\left(2s^2 - 5st - T^2\right) - \left(s^2 + 7st - T^2\right) =$

Introduction

In algebra, simplifying expressions is a crucial skill that helps us solve equations and inequalities. In this article, we will focus on simplifying a given expression involving polynomials. We will use the distributive property, combine like terms, and apply other algebraic techniques to simplify the expression.

The Given Expression

The given expression is:

$$\left(2s^2 - 5st - t^2\right) - \left(s^2 + 7st - t^2\right)$$

Our goal is to simplify this expression by combining like terms and applying other algebraic techniques.

Step 1: Distribute the Negative Sign

To simplify the expression, we will start by distributing the negative sign to the terms inside the second set of parentheses.

$$\left(2s^2 - 5st - t^2\right) - \left(s^2 + 7st - t^2\right)$$

$$= 2s^2 - 5st - t^2 - s^2 - 7st + t^2$$

Step 2: Combine Like Terms

Now, we will combine like terms by adding or subtracting the coefficients of the same variables.

$$= 2s^2 - 5st - t^2 - s^2 - 7st + t^2$$

$$= (2s^2 - s^2) + (-5st - 7st) + (-t^2 + t^2)$$

$$= s^2 - 12st$$

Step 3: Simplify the Expression

We have now simplified the expression by combining like terms. The final simplified expression is:

$$s^2 - 12st$$

Conclusion

In this article, we simplified the given expression by distributing the negative sign and combining like terms. We applied the distributive property and algebraic techniques to simplify the expression. The final simplified expression is $s^2 - 12st$. This expression can be used to solve equations and inequalities involving polynomials.

Real-World Applications

Simplifying expressions is a crucial skill in algebra that has many real-world applications. In physics, for example, simplifying expressions is used to solve problems involving motion and energy. In engineering, simplifying expressions is used to design and optimize systems. In economics, simplifying expressions is used to model and analyze economic systems.

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 set of parentheses.
• Combine like terms by adding or subtracting the coefficients of the same variables.
• Use algebraic techniques such as factoring and canceling to simplify the expression.
• Check your work by plugging in values and verifying that the expression is true.

Common Mistakes

Here are some common mistakes to avoid when simplifying expressions:

• Failing to distribute the negative sign to the terms inside the second set of parentheses.
• Failing to combine like terms by adding or subtracting the coefficients of the same variables.
• Failing to use algebraic techniques such as factoring and canceling to simplify the expression.
• Failing to check your work by plugging in values and verifying that the expression is true.

Practice Problems

Here are some practice problems to help you practice simplifying expressions:

• Simplify the expression: $\left(3x^2 + 2xy - y^2\right) - \left(x^2 - 4xy + y^2\right)$
• Simplify the expression: $\left(2x^2 - 5xy + 3y^2\right) + \left(x^2 + 2xy - 4y^2\right)$
• Simplify the expression: $\left(x^2 + 2xy - 3y^2\right) - \left(2x^2 - 5xy + y^2\right)$

Conclusion

Introduction

In our previous article, we simplified the expression $\left(2s^2 - 5st - t^2\right) - \left(s^2 + 7st - t^2\right)$ using the distributive property and combining like terms. In this article, we will answer some frequently asked questions about simplifying expressions.

Q&A

Q: What is the distributive property?

A: The distributive property is a mathematical concept that allows us to distribute a negative sign or a coefficient to the terms inside a set of parentheses.

Q: How do I apply the distributive property?

A: To apply the distributive property, simply distribute the negative sign or the coefficient to the terms inside the set of parentheses. For example, if we have the expression $\left(2s^2 - 5st - t^2\right) - \left(s^2 + 7st - t^2\right)$, we would distribute the negative sign to the terms inside the second set of parentheses.

Q: What is the difference between combining like terms and simplifying an expression?

A: Combining like terms involves adding or subtracting the coefficients of the same variables, while simplifying an expression involves using algebraic techniques such as factoring and canceling to simplify the expression.

Q: How do I combine like terms?

A: To combine like terms, simply add or subtract the coefficients of the same variables. For example, if we have the expression $2x^2 + 3x^2$, we would combine the like terms by adding the coefficients: $2x^2 + 3x^2 = 5x^2$.

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

A: Some common mistakes to avoid when simplifying expressions include failing to distribute the negative sign, failing to combine like terms, and failing to use algebraic techniques such as factoring and canceling.

Q: How do I check my work when simplifying expressions?

A: To check your work, simply plug in values and verify that the expression is true. For example, if we have the expression $s^2 - 12st$, we can plug in values such as $s = 2$ and $t = 3$ to verify that the expression is true.

Q: What are some real-world applications of simplifying expressions?

A: Simplifying expressions has many real-world applications, including physics, engineering, and economics. In physics, for example, simplifying expressions is used to solve problems involving motion and energy. In engineering, simplifying expressions is used to design and optimize systems. In economics, simplifying expressions is used to model and analyze economic systems.

Q: How can I practice simplifying expressions?

A: You can practice simplifying expressions by working on practice problems, such as the ones listed below:

• Simplify the expression: $\left(3x^2 + 2xy - y^2\right) - \left(x^2 - 4xy + y^2\right)$
• Simplify the expression: $\left(2x^2 - 5xy + 3y^2\right) + \left(x^2 + 2xy - 4y^2\right)$
• Simplify the expression: $\left(x^2 + 2xy - 3y^2\right) - \left(2x^2 - 5xy + y^2\right)$

Conclusion

In conclusion, simplifying expressions is a crucial skill in algebra that has many real-world applications. By following the steps outlined in this article, you can simplify expressions and solve equations and inequalities involving polynomials. Remember to use the distributive property, combine like terms, and apply other algebraic techniques to simplify the expression. With practice and patience, you can become proficient in simplifying expressions and solving problems involving polynomials.

Practice Problems

Here are some practice problems to help you practice simplifying expressions:

• Simplify the expression: $\left(3x^2 + 2xy - y^2\right) - \left(x^2 - 4xy + y^2\right)$
• Simplify the expression: $\left(2x^2 - 5xy + 3y^2\right) + \left(x^2 + 2xy - 4y^2\right)$
• Simplify the expression: $\left(x^2 + 2xy - 3y^2\right) - \left(2x^2 - 5xy + y^2\right)$

Additional Resources

Here are some additional resources to help you learn more about simplifying expressions:

• Khan Academy: Simplifying Expressions
• Mathway: Simplifying Expressions
• IXL: Simplifying Expressions

Conclusion

In conclusion, simplifying expressions is a crucial skill in algebra that has many real-world applications. By following the steps outlined in this article, you can simplify expressions and solve equations and inequalities involving polynomials. Remember to use the distributive property, combine like terms, and apply other algebraic techniques to simplify the expression. With practice and patience, you can become proficient in simplifying expressions and solving problems involving polynomials.