Simplify The Expression: { (2y - 11)(y^2 - 3y + 2)$}$

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Introduction

In algebra, simplifying expressions is a crucial skill that helps us solve equations and inequalities. One of the most common methods of simplifying expressions is by using the distributive property, which states that for any real numbers a, b, and c, a(b + c) = ab + ac. In this article, we will use the distributive property to simplify the given expression $(2y - 11)(y^2 - 3y + 2)$.

Understanding the Expression

The given expression is a product of two binomials, $(2y - 11)$ and $(y^2 - 3y + 2)$. To simplify this expression, we need to multiply each term in the first binomial with each term in the second binomial. This process is called the FOIL method, which stands for First, Outer, Inner, Last.

FOIL Method

The FOIL method is a technique used to multiply two binomials. It involves multiplying the first terms of each binomial, then the outer terms, then the inner terms, and finally the last terms. The resulting expression is then simplified by combining like terms.

First Terms

The first terms of each binomial are $2y$ and $y^2$. Multiplying these two terms gives us:

$$2y \cdot y^2 = 2y^3$$

Outer Terms

The outer terms of each binomial are $2y$ and $-3y$. Multiplying these two terms gives us:

$$2y \cdot -3y = -6y^2$$

Inner Terms

The inner terms of each binomial are $-11$ and $-3y$. Multiplying these two terms gives us:

$$-11 \cdot -3y = 33y$$

Last Terms

The last terms of each binomial are $-11$ and $2$. Multiplying these two terms gives us:

$$-11 \cdot 2 = -22$$

Combining Like Terms

Now that we have multiplied each term in the first binomial with each term in the second binomial, we can combine like terms to simplify the expression. The resulting expression is:

$$2y^3 - 6y^2 + 33y - 22$$

Final Answer

The simplified expression is $2y^3 - 6y^2 + 33y - 22$.

Conclusion

In this article, we used the distributive property and the FOIL method to simplify the expression $(2y - 11)(y^2 - 3y + 2)$. We multiplied each term in the first binomial with each term in the second binomial and then combined like terms to simplify the expression. The resulting expression is $2y^3 - 6y^2 + 33y - 22$.

Tips and Tricks

• When simplifying expressions, always use the distributive property and the FOIL method.
• Make sure to combine like terms to simplify the expression.
• Use the FOIL method to multiply two binomials.

Common Mistakes

• Failing to use the distributive property and the FOIL method.
• Not combining like terms.
• Making errors when multiplying terms.

Real-World Applications

• Simplifying expressions is a crucial skill in algebra and is used in many real-world applications, such as physics, engineering, and computer science.
• The distributive property and the FOIL method are used to simplify expressions in many mathematical problems.

Further Reading

• For more information on simplifying expressions, see the article on "Simplifying Expressions: A Step-by-Step Guide".
• For more information on the distributive property and the FOIL method, see the article on "The Distributive Property and the FOIL Method: A Guide".

References

• [1] "Algebra" by Michael Artin, 2nd edition, Prentice Hall, 2011.
• [2] "Mathematics: A Human Approach" by Harold R. Jacobs, 2nd edition, W.H. Freeman and Company, 1987.

Glossary

• Binomial: A polynomial with two terms.
• Distributive Property: A property of real numbers that states that for any real numbers a, b, and c, a(b + c) = ab + ac.
• FOIL Method: A technique used to multiply two binomials.
• Like Terms: Terms that have the same variable and exponent.
  

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Introduction

In our previous article, we used the distributive property and the FOIL method to simplify the expression $(2y - 11)(y^2 - 3y + 2)$. In this article, we will answer some frequently asked questions about simplifying expressions and provide additional tips and tricks.

Q&A

Q: What is the distributive property?

A: The distributive property is a property of real numbers that states that for any real numbers a, b, and c, a(b + c) = ab + ac. This means that we can multiply a single term by two or more terms inside a set of parentheses.

Q: What is the FOIL method?

A: The FOIL method is a technique used to multiply two binomials. It involves multiplying the first terms of each binomial, then the outer terms, then the inner terms, and finally the last terms.

Q: How do I simplify an expression using the distributive property and the FOIL method?

A: To simplify an expression using the distributive property and the FOIL method, follow these steps:

1. Multiply each term in the first binomial with each term in the second binomial.
2. Combine like terms to simplify the expression.

Q: What are like terms?

A: Like terms are terms that have the same variable and exponent. For example, 2x and 4x are like terms because they both have the variable x and the same exponent (1).

Q: How do I know when to use the distributive property and the FOIL method?

A: You should use the distributive property and the FOIL method whenever you need to multiply two or more terms inside a set of parentheses.

Q: Can I use the distributive property and the FOIL method to simplify expressions with more than two binomials?

A: Yes, you can use the distributive property and the FOIL method to simplify expressions with more than two binomials. However, it may be more complicated and require more steps.

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

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

• Failing to use the distributive property and the FOIL method
• Not combining like terms
• Making errors when multiplying terms

Tips and Tricks

• Always use the distributive property and the FOIL method when simplifying expressions.
• Make sure to combine like terms to simplify the expression.
• Use the FOIL method to multiply two binomials.
• Check your work by plugging in values for the variables.

Real-World Applications

• Simplifying expressions is a crucial skill in algebra and is used in many real-world applications, such as physics, engineering, and computer science.
• The distributive property and the FOIL method are used to simplify expressions in many mathematical problems.

Further Reading

• For more information on simplifying expressions, see the article on "Simplifying Expressions: A Step-by-Step Guide".
• For more information on the distributive property and the FOIL method, see the article on "The Distributive Property and the FOIL Method: A Guide".

References

• [1] "Algebra" by Michael Artin, 2nd edition, Prentice Hall, 2011.
• [2] "Mathematics: A Human Approach" by Harold R. Jacobs, 2nd edition, W.H. Freeman and Company, 1987.

Glossary

• Binomial: A polynomial with two terms.
• Distributive Property: A property of real numbers that states that for any real numbers a, b, and c, a(b + c) = ab + ac.
• FOIL Method: A technique used to multiply two binomials.
• Like Terms: Terms that have the same variable and exponent.

Additional Resources

• Khan Academy: Simplifying Expressions
• Mathway: Simplifying Expressions
• Wolfram Alpha: Simplifying Expressions

Conclusion

Simplifying expressions is a crucial skill in algebra and is used in many real-world applications. The distributive property and the FOIL method are essential tools for simplifying expressions. By following the steps outlined in this article and using the tips and tricks provided, you can become proficient in simplifying expressions and tackle even the most complex mathematical problems.