Simplify The Expression:${ (3x - 2)(x - 3) }$

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Introduction

In algebra, simplifying expressions is a crucial skill that helps in solving equations and inequalities. It involves combining like terms, removing parentheses, and rearranging the expression to make it easier to work with. In this article, we will focus on simplifying the expression (3x - 2)(x - 3) using the distributive property and combining like terms.

Understanding the Distributive Property

The distributive property is a fundamental concept in algebra that allows us to expand expressions by multiplying each term inside the parentheses with the term outside the parentheses. In the expression (3x - 2)(x - 3), we have two binomials (expressions with two terms) that need to be multiplied together. To simplify this expression, we will use the distributive property to expand it.

Expanding the Expression

To expand the expression (3x - 2)(x - 3), we will multiply each term inside the first parentheses with each term inside the second parentheses. This can be done by multiplying the first term of the first parentheses (3x) with the first term of the second parentheses (x), then multiplying the first term of the first parentheses (3x) with the second term of the second parentheses (-3), and so on.

Using the distributive property, we get:

(3x - 2)(x - 3) = 3x(x) + 3x(-3) - 2(x) - 2(-3)

Simplifying the Expression

Now that we have expanded the expression, we can simplify it by combining like terms. Like terms are terms that have the same variable raised to the same power. In this case, we have the following like terms:

  • 3x(x) = 3x^2
  • 3x(-3) = -9x
  • -2(x) = -2x
  • -2(-3) = 6

Combining these like terms, we get:

3x^2 - 9x - 2x + 6

Combining Like Terms

Now that we have identified the like terms, we can combine them by adding or subtracting their coefficients. In this case, we have two like terms (-9x and -2x) that can be combined by adding their coefficients.

-9x - 2x = -11x

So, the simplified expression becomes:

3x^2 - 11x + 6

Conclusion

In this article, we simplified the expression (3x - 2)(x - 3) using the distributive property and combining like terms. We expanded the expression by multiplying each term inside the first parentheses with each term inside the second parentheses, and then simplified it by combining like terms. The final simplified expression is 3x^2 - 11x + 6.

Tips and Tricks

  • When simplifying expressions, always look for like terms and combine them by adding or subtracting their coefficients.
  • Use the distributive property to expand expressions by multiplying each term inside the parentheses with the term outside the parentheses.
  • Be careful when multiplying negative numbers, as it can result in a positive product.

Common Mistakes to Avoid

  • Failing to identify like terms and combine them.
  • Not using the distributive property to expand expressions.
  • Making mistakes when multiplying negative numbers.

Real-World Applications

Simplifying expressions is a crucial skill that has many real-world applications. In physics, for example, simplifying expressions is used to solve equations of motion and energy. In engineering, simplifying expressions is used to design and optimize systems. In finance, simplifying expressions is used to calculate interest rates and investment returns.

Final Thoughts

Simplifying expressions is a fundamental skill in algebra that helps in solving equations and inequalities. By using the distributive property and combining like terms, we can simplify expressions and make them easier to work with. In this article, we simplified the expression (3x - 2)(x - 3) and provided tips and tricks for simplifying expressions. We also discussed common mistakes to avoid and real-world applications of simplifying expressions.

Introduction

In our previous article, we simplified the expression (3x - 2)(x - 3) using the distributive property and combining like terms. In this article, we will answer some frequently asked questions (FAQs) related to simplifying expressions.

Q&A

Q: What is the distributive property?

A: The distributive property is a fundamental concept in algebra that allows us to expand expressions by multiplying each term inside the parentheses with the term outside the parentheses.

Q: How do I identify like terms?

A: Like terms are terms that have the same variable raised to the same power. To identify like terms, look for terms that have the same variable and exponent.

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

A: Combining like terms involves adding or subtracting the coefficients of like terms, while simplifying expressions involves using the distributive property and combining like terms to make the expression easier to work with.

Q: Can I simplify expressions with variables in the denominator?

A: Yes, you can simplify expressions with variables in the denominator. However, you need to be careful when multiplying and dividing fractions.

Q: How do I simplify expressions with exponents?

A: To simplify expressions with exponents, use the rules of exponents, such as multiplying exponents when multiplying like bases and adding exponents when adding like bases.

Q: Can I simplify expressions with absolute values?

A: Yes, you can simplify expressions with absolute values. However, you need to be careful when dealing with negative numbers and absolute values.

Q: How do I simplify expressions with fractions?

A: To simplify expressions with fractions, use the rules of fractions, such as multiplying fractions by multiplying the numerators and denominators and dividing fractions by inverting the second fraction and multiplying.

Q: Can I simplify expressions with radicals?

A: Yes, you can simplify expressions with radicals. However, you need to be careful when dealing with square roots and other radicals.

Q: How do I simplify expressions with multiple variables?

A: To simplify expressions with multiple variables, use the distributive property and combine like terms to make the expression easier to work with.

Q: Can I simplify expressions with trigonometric functions?

A: Yes, you can simplify expressions with trigonometric functions. However, you need to be familiar with the rules of trigonometry and the properties of trigonometric functions.

Q: How do I simplify expressions with logarithmic functions?

A: To simplify expressions with logarithmic functions, use the rules of logarithms, such as the product rule and the quotient rule.

Tips and Tricks

  • Always look for like terms and combine them by adding or subtracting their coefficients.
  • Use the distributive property to expand expressions by multiplying each term inside the parentheses with the term outside the parentheses.
  • Be careful when dealing with negative numbers, absolute values, fractions, and radicals.
  • Use the rules of exponents, fractions, and logarithms to simplify expressions.

Common Mistakes to Avoid

  • Failing to identify like terms and combine them.
  • Not using the distributive property to expand expressions.
  • Making mistakes when multiplying and dividing fractions.
  • Not being careful when dealing with negative numbers, absolute values, and radicals.

Real-World Applications

Simplifying expressions is a crucial skill that has many real-world applications. In physics, for example, simplifying expressions is used to solve equations of motion and energy. In engineering, simplifying expressions is used to design and optimize systems. In finance, simplifying expressions is used to calculate interest rates and investment returns.

Final Thoughts

Simplifying expressions is a fundamental skill in algebra that helps in solving equations and inequalities. By using the distributive property and combining like terms, we can simplify expressions and make them easier to work with. In this article, we answered some frequently asked questions (FAQs) related to simplifying expressions and provided tips and tricks for simplifying expressions. We also discussed common mistakes to avoid and real-world applications of simplifying expressions.