Problem 46 - Algebra A Combined Function

Question

Add or subtract as indicated. $$ \left(-m^{2}+3\right)-\left(m^{2}-13\right)+\left(6 m^{2}-m+1\right) $$

Step-by-Step Solution

Answer

The simplified expression is \(4m^{2} - m + 17\).

Step 1: Identify and Simplify the Expression

The given expression is \(-m^{2}+3-(m^{2}-13)+(6m^{2}-m+1)\). First, let's distribute any negative signs in the equation. This leads to \(-m^{2}+3-m^{2}+13+6m^{2}-m+1\).

Step 2: Combine Like Terms

Next, we need to combine like terms by organizing the expression into similar categories: \(-m^{2}-m^{2}+6m^{2}\) (for the quadratic terms), and \(3+13+1\) (for the constant terms), and the single term \(-m\) (for the linear term).

Step 3: Solve for Quadratic Terms

Add the quadratic terms together: \(-m^{2}-m^{2}+6m^{2} = 4m^{2}\).

Step 4: Solve for the Linear Term

The linear term is just \(-m\) so it remains unchanged in the simplification process.

Step 5: Solve for Constant Terms

Add the constant terms: \(3 + 13 + 1 = 17\).

Step 6: Write the Final Simplified Expression

Now, combine the results from each category: \(4m^{2} - m + 17\). This is the fully simplified expression.

Key Concepts

Algebraic Simplification

Algebraic simplification is the process of rewriting an algebraic expression in its simplest form. This involves condensing the expression by combining like terms and eliminating any unnecessary components. Simplifying makes an expression easier to understand and work with.

In our problem, we begin by simplifying the given expression. The initial step involves distributing negative signs across grouping symbols. This is crucial because it helps accurately represent terms with correct signs. After this distribution, we have a straightforward list of terms to work with. By breaking them down, you simplify the task of further combining them.

Always look out for negative signs before terms and inside parentheses. Simplification helps to manage calculations efficiently, as we can eliminate redundant elements and organize like terms.

Like Terms

Like terms are terms that contain the same variable raised to the same power. In polynomial expressions, identifying and combining like terms is essential to simplification. Like terms share the same variable component, even if their coefficients are different.

For instance, in the original exercise, we identified quadratic terms as

• \(-m^2\), \(-m^2\), and \(6m^2\).

These terms all have the variable \(m\) raised to the power of 2, which makes them like terms. By combining their coefficients, we simplify it to \(4m^2\).

Remember:

• Combine the coefficients of like terms by addition or subtraction.
• Keep the variable and its exponent unchanged.

Being able to group and simplify like terms is a foundational mathematical skill that aids in the further simplification of expressions or solving equations.

Quadratic Expressions

A quadratic expression is a type of polynomial expression where the highest power of the variable is two. These expressions typically take the form of \(ax^2 + bx + c\). It's important to understand how to simplify and work with quadratic expressions because they appear frequently in algebra and various applications such as physics and engineering.

In the simplified solution of our problem, the final quadratic expression features a term with \(m^2\) as its highest power: \(4m^2 - m + 17\). Here, \(4m^2\) is the quadratic term, \(-m\) is the linear term, and \(17\) is the constant term.

Key points about quadratic expressions:

• The quadratic term \(ax^2\) defines the shape of the graph of the expression when plotted.
• Quadratic expressions can be factored or used within equations to find "roots" or solutions.
• Understanding the structure helps in solving quadratic equations and performing operations, like completing the square or using the quadratic formula.

Recognizing and working with quadratic expressions is essential for progressing in algebra and solving more complicated mathematical problems. It provides insight into the behavior and intersection points of various functions.