Simplifying $-3(y-5)^2-9+7y$ A Step-by-Step Guide

In the realm of mathematics, simplifying expressions is a fundamental skill. It allows us to rewrite complex expressions in a more manageable and understandable form. In this article, we will delve into the process of simplifying the expression $-3(y-5)^2-9+7y$, meticulously examining each step and identifying the correct statements about the simplification process and the resulting product. This comprehensive guide aims to provide a clear understanding of the order of operations, the distributive property, and combining like terms, all crucial concepts in algebraic simplification.

Understanding the Expression

Before we embark on the simplification journey, let's first dissect the expression $-3(y-5)^2-9+7y$. This expression comprises several components: a squared binomial $(y-5)^2$, a constant term -9, and a linear term $7y$. The presence of the squared binomial and the multiplication by -3 necessitate a specific order of operations to ensure accurate simplification. The key to successfully simplifying this expression lies in adhering to the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This order dictates the sequence in which we perform the mathematical operations.

Order of Operations (PEMDAS/BODMAS)

To simplify any mathematical expression correctly, it's crucial to follow the order of operations. This order ensures that everyone arrives at the same simplified form. PEMDAS stands for:

• Parentheses (or Brackets)
• Exponents (or Orders)
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

Following this order, we first address any operations within parentheses, then exponents, followed by multiplication and division (from left to right), and finally, addition and subtraction (also from left to right).

Step-by-Step Simplification

Now, let's apply the order of operations to simplify the expression $-3(y-5)^2-9+7y$:

1. Expanding the Squared Binomial

The first step, according to PEMDAS, is to address the exponent. We need to expand the squared binomial $(y-5)^2$. This means multiplying the binomial by itself:

$(y-5)^2 = (y-5)(y-5)$

To multiply these binomials, we can use the FOIL method (First, Outer, Inner, Last) or the distributive property:

$(y-5)(y-5) = y(y) + y(-5) - 5(y) - 5(-5)$

$= y^2 - 5y - 5y + 25$

$= y^2 - 10y + 25$

2. Distributing the -3

Now that we've expanded the squared binomial, we substitute it back into the original expression:

$-3(y^2 - 10y + 25) - 9 + 7y$

Next, we distribute the -3 across the trinomial:

$-3(y^2) - 3(-10y) - 3(25) - 9 + 7y$

$= -3y^2 + 30y - 75 - 9 + 7y$

3. Combining Like Terms

The final step is to combine like terms. Like terms are terms that have the same variable raised to the same power. In this expression, we have two terms with the variable 'y' ($30y$ and $7y$) and two constant terms (-75 and -9):

$-3y^2 + 30y + 7y - 75 - 9$

Combining these terms, we get:

$-3y^2 + 37y - 84$

Therefore, the simplified expression is $-3y^2 + 37y - 84$.

Analyzing the Statements

Now that we have simplified the expression, let's evaluate the statements provided:

A. The first step in simplifying is to distribute the -3 throughout the parentheses.

This statement is incorrect. As we saw in the step-by-step simplification, the first step is to expand the squared binomial $(y-5)^2$ due to the exponent. The order of operations dictates that we address exponents before multiplication (distribution).

B. There are 3 terms in the simplified product.

Let's examine the simplified expression: $-3y^2 + 37y - 84$. We can clearly identify three distinct terms: $-3y^2$, $37y$, and $-84$. Therefore, this statement is correct.

Key Concepts Revisited

Let's reinforce the key concepts we've utilized in simplifying this expression:

• Order of Operations (PEMDAS/BODMAS): The foundation of simplifying any mathematical expression. It ensures consistent results by establishing a specific sequence for performing operations.
• Expanding Squared Binomials: Mastering the expansion of binomials, particularly squared binomials, is crucial in algebra. Techniques like the FOIL method or the distributive property are invaluable tools.
• Distributive Property: This property allows us to multiply a single term by a group of terms within parentheses. It's a fundamental technique for expanding expressions.
• Combining Like Terms: Identifying and combining like terms is essential for simplifying expressions. It involves adding or subtracting terms that have the same variable raised to the same power.

Common Pitfalls to Avoid

When simplifying expressions, it's easy to make mistakes. Here are some common pitfalls to watch out for:

• Ignoring the Order of Operations: Deviating from PEMDAS can lead to incorrect results. Always adhere to the established order.
• Incorrectly Expanding Binomials: Errors in applying the FOIL method or distributive property can lead to an incorrect expansion.
• Distributing Negatives: When distributing a negative number, remember to apply the negative sign to every term inside the parentheses.
• Combining Unlike Terms: Only combine terms that have the same variable raised to the same power.

Conclusion

Simplifying algebraic expressions is a cornerstone of mathematics. By understanding and applying the order of operations, the distributive property, and the technique of combining like terms, we can effectively reduce complex expressions to their simplest forms. In this article, we have meticulously simplified the expression $-3(y-5)^2-9+7y$, highlighting each step and emphasizing the importance of adhering to mathematical principles. Remember, practice is key to mastering these skills. By working through various examples and paying attention to detail, you can confidently simplify a wide range of algebraic expressions.

In conclusion, remember that attention to detail and a solid understanding of mathematical principles are essential for success in simplifying expressions. By mastering these skills, you'll not only be able to tackle complex mathematical problems but also develop a deeper appreciation for the elegance and precision of mathematics.