Problem 10 - Algebra 2 and Trigonometry

Question

In \(3-12,\) write the sum or difference of the given polynomials in simplest form. $$ \left(4 x^{2}-3 x-5\right)-\left(3 x^{2}-10 x+3\right) $$

Step-by-Step Solution

Answer

The simplest form is \( x^2 + 7x - 8 \).

Step 1: Distribute the Negative Sign

First, distribute the negative sign across the polynomial in the parentheses on the right side of the expression. \[-(3x^2 - 10x + 3) = -3x^2 + 10x - 3\]

Step 2: Write Down the Expression

Substitute the distributed terms into the expression, yielding: \[(4x^2 - 3x - 5) - (3x^2 - 10x + 3) = 4x^2 - 3x - 5 - 3x^2 + 10x - 3\]

Step 3: Combine Like Terms

Group and combine like terms in the expression.- Combine the \( x^2 \) terms: \[4x^2 - 3x^2 = x^2\]- Combine the \( x \) terms: \[-3x + 10x = 7x\]- Combine the constant terms: \[-5 - 3 = -8\]The simplest form of the expression is:\[x^2 + 7x - 8\]

Key Concepts

Distributing Negative Sign

When working with polynomial subtraction, it's essential to manage negative signs properly. This step ensures that each term within the subtracted polynomial is correctly adjusted before further simplification. To do this, we distribute the negative sign across every term of the polynomial in the parentheses.

For example, consider the expression:

• The original expression is: \[(4x^2 - 3x - 5) - (3x^2 - 10x + 3)\]
• Distribute the negative: \[-(3x^2 - 10x + 3) = -3x^2 + 10x - 3\]

Notice how each term inside the parentheses changes its sign. The positive 3xyields a negative 3x², negative 10x transforms into positive 10x, and positive 3 becomes -3.

By distributing accurately, you set the foundation for the next step, making sure the problem won't have errors later on.

Combining Like Terms

Once the negative sign is distributed, the next step in simplifying the expression involves combining like terms. This step is straightforward if done systematically, and it involves grouping the terms with the same variable power together. Here’s how you can do that:

• Start with rearranging the expression post distribution, which is: \[4x^2 - 3x - 5 - 3x^2 + 10x - 3\]
• Combine the \(x^2\) terms: \[4x^2 - 3x^2 = x^2\]
• Next, handle the \(x\) terms: \[-3x + 10x = 7x\]
• Finally, look at the constant terms: \[-5 - 3 = -8\]

Grouping terms by their type and then performing the operations allows for a much tidier and understandable expression. Always double-check each grouping, as skipping or miscalculating can create errors in the final form.

Simplified Expression

After combining like terms, you arrive at the expression in its simplest form. Simplifying expressions not only helps in easier calculations but also in understanding the structure of the polynomial better.

The process so far has led us to:

• From the combined terms: \[x^2 + 7x - 8\]

This final expression, \(x^2 + 7x - 8\), is both simplified and the most concise form of the original polynomial subtraction problem.

Remember that a well-simplified expression should not have any like terms left to combine or any unnecessary parentheses. This makes working with the expression in any further calculations as efficient as it can be.