Problem 25 - Algebra A Combined Function

Question

Subtract. $$ (5 x+8)-\left(-2 x^{2}-6 x+8\right) $$

Step-by-Step Solution

Answer

The solution is \(2x^2 + 11x\).

Step 1: Distribute the Negative Sign

The problem involves subtracting the entire expression \(-2x^2 - 6x + 8\) from \(5x+8\). To simplify, distribute the negative sign across the second expression: \[ (5x + 8) - (-2x^2 - 6x + 8) = 5x + 8 + 2x^2 + 6x - 8 \]

Step 2: Combine Like Terms

Now that we have simplified, the next step is to combine like terms. We have the terms \(+2x^2\), \(5x\ + 6x\), and \(8 - 8\). Combine like terms: \[ 2x^2 + (5x + 6x) + (8 - 8) = 2x^2 + 11x + 0 \]

Step 3: Write the Final Expression

After combining like terms, we are left with the simplified and final expression: \[ 2x^2 + 11x \]

Key Concepts

Distributing Negatives

When solving problems that involve subtracting expressions, distributing negatives is an important step that helps to simplify the problem. Consider the operation in algebra where subtraction involves not just removing something, but also adding the opposite. This is akin to distributing a negative sign across terms inside parentheses, effectively flipping their signs.
Imagine a problem like \[ (5x + 8) - (-2x^2 - 6x + 8). \]
Here, you want to move the negative sign through the expression - Change \(-2x^2\) to \(+2x^2\).- Change \(-6x\) to \(+6x\).- Change \(+8\) to \(-8\).So, after distributing, the operation looks like:\[ 5x + 8 + 2x^2 + 6x - 8. \]
This step is crucial because it sets the stage for combining like terms, making it easier to see which terms can be grouped together.

Combining Like Terms

Once the expressions are simplified by distributing negatives, the next key concept is combining like terms. Like terms are terms whose variables (and their exponents, if applicable) are the same. This makes it possible to "combine" or simplify them into a single term with the same variable.
In the expression:\[ 2x^2 + 5x + 6x - 8 + 8, \]
we identify terms that are alike:- \(2x^2\) is standalone because there's no other \(x^2\) term.- Combine \(5x\) and \(6x\), resulting in \(11x\).- The constants \(+8\) and \(-8\) cancel each other out.After combining, we are left with a cleaner, simplified expression:\[ 2x^2 + 11x. \]
This allows further operations to be carried out more easily, as the equation is now neatly packed into fewer terms.

Polynomial Subtraction

Polynomial subtraction involves careful manipulation to simplify expressions involving multiple powers of variables. Consider polynomial subtraction as removing each individual component of one polynomial from the related parts of another polynomial.
In our example, the task starts by identifying the polynomial to be subtracted. Initially, that's \(-2x^2 - 6x + 8\) from \(5x+8\). By applying the distribution of the negative sign, you effectively turn the subtraction into addition of the opposite:\[ (5x + 8) + (2x^2 + 6x - 8). \]
This transformation changes the initial subtraction problem into an addition of components with modified signs. After distribution comes combining like terms, simplifying the structure to:\[ 2x^2 + 11x. \]Remember, each step in polynomial subtraction, including distributing negatives and combining like terms, is designed to streamline and simplify until you reach the most concise form possible.