Problem 8 - Algebra A Combined Function

Question

Perform each division. See Examples 1 through 3. $$ \frac{14 m^{2}-27 m^{3}}{7 m} $$

Step-by-Step Solution

Answer

The result is \( 2m - \frac{27}{7}m^2 \).

Step 1: Break Down the Expression

We start by rewriting the given expression \( \frac{14 m^{2}-27 m^{3}}{7 m} \) by separating the numerator. The numerator can be split as \( 14m^2 \) and \(-27m^3 \).

Step 2: Divide Each Term Separately

Now divide each term in the numerator by the denominator \( 7m \). This gives us two separate expressions: \( \frac{14m^2}{7m} \) and \( -\frac{27m^3}{7m} \).

Step 3: Simplify Each Division

For \( \frac{14m^2}{7m} \), divide the coefficients and subtract the exponents of \( m \). This results in \( 2m^{2-1} = 2m \). For \( -\frac{27m^3}{7m} \), divide the coefficients to get \( -\frac{27}{7} \) and subtract the exponents of \( m \), which results in \( m^{3-1} = m^2 \). The term is \( -\frac{27}{7}m^2 \).

Step 4: Combine the Results

Combine the simplified terms from Step 3 to get the final result: \( 2m - \frac{27}{7}m^2 \).

Key Concepts

Polynomial division

Polynomial division is an essential technique used in algebra to divide polynomials by other polynomials or monomials. When faced with a polynomial like \( \frac{14m^2 - 27m^3}{7m} \), the division process involves distributing the division over each term in the polynomial, which simplifies the entire process.

Here is how you can do it effectively:

• First, separate the terms in the numerator and designate the denominator to each of these terms. For our example, divide \( 14m^2 \) and \( -27m^3 \) separately by \( 7m \).
• The goal is to simplify each term individually, which is more manageable than attempting to simplify the whole polynomial expression at once.
• Combining the results from each division gives you the final simplified polynomial expression, which is \( 2m - \frac{27}{7}m^2 \).

This method maintains accuracy and clarity in solving polynomial division problems and lays the groundwork for more advanced algebraic techniques.

Simplifying expressions

Simplifying expressions in algebra involves reducing expressions to their simplest form. This process often includes dealing with coefficients, variables, and exponents. It’s about making the expression more straightforward and workable.

For example, with the expression \( \frac{14m^2 - 27m^3}{7m} \), the simplification process begins by separating each term, employing basic arithmetic, and exponent laws:

• Take \( \frac{14m^2}{7m} \): Simplify by dividing the coefficients (14 by 7) to get 2, and subtract the exponent of the variable \( m^2 - m = m^{2-1} = m \), thus resulting in \( 2m \).
• Next, the term \( -\frac{27m^3}{7m} \): Break it down by dividing the coefficients to get \( -\frac{27}{7} \), and perform the subtraction of exponents \( m^{3-1} = m^2 \), resulting in \( -\frac{27}{7}m^2 \).

Once you have simplified each component, combine them to complete the expression, yielding \( 2m - \frac{27}{7}m^2 \). Simplifying makes solving equations quicker and helps understand the behavior of polynomial functions.

Exponents subtraction

The concept of exponents subtraction is crucial when dividing terms with exponents. This rule states that when you divide powers with the same base, you subtract the exponents. It's a straightforward yet powerful arithmetic rule that simplifies complex algebraic expressions.

Here's how it works with our example:

• For the term \( \frac{14m^2}{7m} \), since the bases are identical \( m \), subtract the exponent \( 2-1 \) to obtain \( m^{1} \).
• Similarly, the term \( -\frac{27m^3}{7m} \) follows the same rule: subtract the exponents \( 3-1 \) to get \( m^2 \).

Remember that subtracting exponents only applies when the bases are the same, which maintains uniformity and simplicity in algebraic operations.

Practicing exponent subtraction helps in unraveling more detailed algebraic problems and enhances overall mathematical fluency.