Problem 40 - Algebra 2

Question

Simplify \(\left(c^{2}-6 c d-2 d^{2}\right)+\left(7 c^{2}-c d+8 d^{2}\right)-\left(-c^{2}+5 c d-d^{2}\right)\)

Step-by-Step Solution

Answer

The simplified expression is \(9c^2 - 12cd + 7d^2\).

Step 1: Distribute the Negative Sign

The expression to simplify is \( \left(c^2 - 6cd - 2d^2\right) + \left(7c^2 - cd + 8d^2\right) - \left(-c^2 + 5cd - d^2\right) \). First, distribute the negative sign across the last term: \(-\left(-c^2 + 5cd - d^2\right) = c^2 - 5cd + d^2\). Substitute to get: \( \left(c^2 - 6cd - 2d^2\right) + \left(7c^2 - cd + 8d^2\right) + \left(c^2 - 5cd + d^2\right) \).

Step 2: Combine Like Terms

Now combine like terms in the expression: \(c^2, 7c^2,\) and \(c^2\) give \(9c^2\). \(-6cd, -cd,\) and \(-5cd\) give \(-12cd\). \(-2d^2, 8d^2,\) and \(d^2\) give \(7d^2\).

Step 3: Write the Simplified Expression

Combine these results to get the simplified expression: \(9c^2 - 12cd + 7d^2\).

Key Concepts

Distributing Negative Signs in Polynomial Expressions

When simplifying polynomial expressions, it's essential to manage any negative signs meticulously. This is especially true when these signs are situated just before parentheses. The job here is to distribute the negative sign across each term within the parentheses.
Here's how you can do it:

• Identify the expression within the parentheses that follows a negative sign. For instance, consider the expression \((-(-c^2 + 5cd - d^2))\).
• Apply the negative sign to each term inside the parentheses individually, which changes each sign: \(-(-c^2 + 5cd - d^2) = c^2 - 5cd + d^2\).

Once you have distributed the negative sign, it is important to rewrite the entire expression correctly. This step helps prevent mistakes in further calculations. Distributing these signs correctly ensures that you maintain the integrity of the values as you proceed to combine like terms.

Combining Like Terms for Clarity

Combining like terms is key to simplifying polynomial expressions. It involves identifying and merging terms that share the same variable and power. This process helps in reducing the complexity of an expression, making it easier to manage with fewer terms.
Here's a simple guide to combining like terms:

• Look for terms that have the same variable raised to the same power. For example, locate terms like \((-6cd)\), \((-cd)\), and \((-5cd)\).
• Combine these by adding or subtracting their coefficients, resulting in \-12cd\.

After combining, ensure to record each group accurately in the final expression. Remember, combining like terms makes your calculations more straightforward and helps everyone understand the simplified expression.

Mastering Expression Simplification

Expression simplification aims to condense a mathematical expression to its most straightforward form. Simplification can involve several steps, including distributing negative signs and combining like terms. These steps make the expression clearer and easier to interpret.
The crucial steps in simplification include:

• Accurately follow the order of operations, starting with distributing any negative signs.
• Combine all like terms using addition or subtraction.
• Ensure the expression is written neatly and correctly after simplification: \((9c^2 - 12cd + 7d^2)\).

By breaking down each task into manageable steps, expression simplification becomes straightforward. It's an essential skill for algebra students, preparing for more complex mathematics tasks ahead. With practice, simplifying expressions not only boosts confidence but also enhances problem-solving skills.