Problem 52 - Elementary Algebra

Question

For exercises 49-52, simplify. $$ \frac{m^{3}}{m^{2}+12 m+27}+\frac{27}{m^{2}+12 m+27} $$

Step-by-Step Solution

Answer

\[ \frac{m^2 - 3m + 9}{m + 9} \]

Step 1: Identify Common Denominator

Notice that both fractions have the same denominator: \[ m^2 + 12m + 27 \].

Step 2: Combine the Numerators

Since the denominators are the same, combine the numerators: \[ \frac{m^3 + 27}{m^2 + 12m + 27} \]

Step 3: Factor the Numerator

Factor the numerator, which is a sum of cubes: \[ m^3 + 27 = (m + 3)(m^2 - 3m + 9) \]

Step 4: Factor the Denominator

Factor the denominator: \[ m^2 + 12m + 27 = (m + 3)(m + 9) \]

Step 5: Simplify the Expression

Rewrite the expression with the factors and cancel the common factor \ (m + 3): \[ \frac{(m + 3)(m^2 - 3m + 9)}{(m + 3)(m + 9)} \].Canceling out \ (m + 3), we get: \[ \frac{m^2 - 3m + 9}{m + 9} \]

Key Concepts

Common Denominator

To simplify algebraic expressions involving fractions, the first step often involves finding a common denominator. A common denominator is a shared denominator between fractions, which allows us to combine the fractions easily.
In our problem, the given fractions both have:

• The same denominator: \b \b
• This checks off the first step and ensures the denominators align perfectly.

As the two fractions in question share the denominator \( m^2 + 12m + 27 \), we can proceed by focusing on their numerators next.

Sum of Cubes

The sum of cubes is a special type of polynomial factoring that you need to recognize. Generally, an expression like \( a^3 + b^3 \) can be factored into \( (a + b)(a^2 - ab + b^2) \).
In our problem, the numerator \( m^3 + 27 \) is actually the sum of cubes:

• Identify: \( m \to m \) and \( b = 3 \) since \( 27 = 3^3 \)
• Apply the formula: \( (m + 3)(m^2 - 3m + 9) \)

In this way, we factored the sum of cubes and obtained its simpler form. It’s crucial to memorize the sum of cubes factoring formula to handle such problems efficiently.

Factoring Polynomials

Factoring polynomials is another key step in simplifying algebraic expressions. This process involves breaking down a polynomial into simpler 'factor' polynomials whose product is the original polynomial.
Looking at the denominator in our problem: \( m^2 + 12m + 27 \), we proceed by:

• Finding numbers that multiply to 27 and add up to 12. These numbers are 3 and 9
• Expressing the polynomial as: \( (m + 3)(m + 9) \)

Thus, factoring polynomials helps convert complex expressions into simpler products, making it easier to cancel out common terms in the fraction.

Canceling Common Factors

Canceling common factors simplifies fractions. When a numerator and denominator share a common polynomial factor, they can be canceled out.
For our fraction, we have \( \frac{(m + 3)(m^2 - 3m + 9)}{(m + 3)(m + 9)} \):

• The common factor is \( (m + 3) \)
• We cancel it out from both the numerator and the denominator.

This leaves us with a simplified form of the expression: \( \frac{m^2 - 3m + 9}{m + 9} \).Remember, canceling common factors helps reduce fractions to their simplest forms, eliminating extraneous complexity.