Problem 11 - Algebra A Combined Function

Question

$$ \left(\frac{3}{4} m^{2}-\frac{2}{5} m+\frac{1}{8}\right)+\left(-\frac{1}{4} m^{2}-\frac{3}{10} m+\frac{11}{16}\right) $$

Step-by-Step Solution

Answer

\( \frac{1}{2} m^2 - \frac{7}{10} m + \frac{13}{16} \)

Step 1: Identify Like Terms

Before solving the expression, let's group together like terms: terms with \( m^2 \), terms with \( m \), and constant terms. The expression becomes: \[\left(\frac{3}{4} m^2 - \frac{1}{4} m^2\right) + \left(-\frac{2}{5} m - \frac{3}{10} m\right) + \left(\frac{1}{8} + \frac{11}{16}\right)\]

Step 2: Combine \( m^2 \) Terms

Now, let's combine the \( m^2 \) terms: \[ \frac{3}{4} m^2 - \frac{1}{4} m^2 = \left(\frac{3}{4} - \frac{1}{4}\right) m^2 = \frac{2}{4} m^2 = \frac{1}{2} m^2 \]

Step 3: Combine \( m \) Terms

Next, combine the terms with \( m \): Convert \(-\frac{2}{5} m\) and \(-\frac{3}{10} m\) to have a common denominator, in this case, 10: \(-\frac{4}{10} m - \frac{3}{10} m\) results in: \[ -\frac{4}{10} m - \frac{3}{10} m = -\frac{7}{10} m \]

Step 4: Combine Constant Terms

Now, let's combine the constant terms \( \frac{1}{8} \) and \( \frac{11}{16} \): Convert \( \frac{1}{8} \) to have the same denominator of 16: \( \frac{2}{16} + \frac{11}{16} \) results in: \[ \frac{2}{16} + \frac{11}{16} = \frac{13}{16} \]

Step 5: Write the Final Combined Expression

Now, assemble all the combined terms into a single expression: \[ \frac{1}{2} m^2 - \frac{7}{10} m + \frac{13}{16} \]

Key Concepts

Like Terms

When working with polynomials, one of the first key steps is to identify like terms. These are terms that contain the same variable raised to the same power. For example, in our given expression, terms involving \( m^2 \) are like terms, terms involving \( m \) are like terms, and constant numbers (terms without variables) are their own group of like terms.

• Terms with the same variable part can be combined by adding or subtracting their coefficients.
• In the expression \( \left(\frac{3}{4} m^{2}-\frac{2}{5} m+\frac{1}{8}\right)+\left(-\frac{1}{4} m^{2}-\frac{3}{10} m+\frac{11}{16}\right) \), notice the terms \( \frac{3}{4} m^2 \) and \( -\frac{1}{4} m^2 \) are like terms because they both involve \( m^2 \).
• Likewise, terms that include just \( m \) - like \( -\frac{2}{5} m \) and \( -\frac{3}{10} m \) - should be handled together.

By identifying like terms, computations become more straightforward, allowing you to simplify the polynomial efficiently.

Combining Fractions

Combining fractions involves adding or subtracting them to form a single fraction. To combine fractions, especially when dealing with like terms, it is crucial to handle their coefficients, which are often in fractional form in polynomials.

• To add or subtract fractions like \( -\frac{2}{5} \) and \( -\frac{3}{10} \), we first need to change their denominators to be the same.
• The least common multiple of their denominators (5 and 10) is 10, so we convert \( -\frac{2}{5} \) to \( -\frac{4}{10} \).
• Once the denominators are the same, the numerators can be combined: \( -\frac{4}{10} - \frac{3}{10} = -\frac{7}{10} \).

Understanding this method allows for a seamless combination of fractional coefficients, simplifying the polynomial addition process.

Common Denominator

A common denominator is essential when combining fractions, especially with regard to polynomial operations where terms may have fractional coefficients or constants.

• The goal is to match the denominators, so they can be directly added or subtracted.
• Take the example of combining \( \frac{1}{8} \) and \( \frac{11}{16} \): the common denominator here is 16.
• Convert \( \frac{1}{8} \) to \( \frac{2}{16} \) by multiplying both the numerator and denominator by 2, enabling the computation: \( \frac{2}{16} + \frac{11}{16} = \frac{13}{16} \).

Mastering the art of finding a common denominator facilitates operations on polynomials with fractional components, thus simplifying expressions and making polynomial addition more accessible.