Problem 38 - Algebra for College Students

Question

Reduce each rational expression to its lowest terms. $$\frac{27 x^{3}+y^{3}}{6 x+2 y}$$

Step-by-Step Solution

Answer

The reduced form is \(\frac{9x^2 - 3xy + y^2}{2}\).

Step 1: Identify the Given Rational Expression

The given rational expression is \(\frac{27x^3 + y^3}{6x + 2y}\).

Step 2: Factor the Numerator

The numerator can be factorized using the sum of cubes formula: \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\). Here, \(27x^3\) is \((3x)^3\) and \(y^3\) is \(y^3\). So, the factorization will be \((3x + y)((3x)^2 - (3x)y + y^2) = (3x + y)(9x^2 - 3xy + y^2)\). Thus: \(27x^3 + y^3 = (3x + y)(9x^2 - 3xy + y^2)\).

Step 3: Factor the Denominator

Factor out the greatest common factor (GCF) from the denominator. For \(6x + 2y\), the GCF is 2, so: \(6x + 2y = 2(3x + y)\).

Step 4: Simplify the Expression

Now, substitute the factors back into the rational expression and cancel the common factors. \(\frac{(3x + y)(9x^2 - 3xy + y^2)}{2(3x + y)}\). Since \(3x + y\) is a common factor in the numerator and denominator, they cancel each other out, so: \(\frac{(3x + y)(9x^2 - 3xy + y^2)}{2(3x + y)} = \frac{9x^2 - 3xy + y^2}{2}\).

Step 5: Write the Final Answer

The reduced form of the rational expression is \(\frac{9x^2 - 3xy + y^2}{2}\).

Key Concepts

factorization

Factorization is a key concept in simplifying mathematical expressions. It involves breaking down a complex expression into its simpler building blocks, known as factors. This process can make it easier to solve equations, simplify expressions, and perform other mathematical operations.

To factorize a polynomial, look for common factors and use algebraic identities. In our exercise, the numerator was factorized using the sum of cubes formula, \[a^3 + b^3 = (a + b)(a^2 - ab + b^2)\]

Recognizing and applying the correct factorization formula is crucial. This helps to identify patterns and break down the polynomial into simpler expressions.

greatest common factor

The greatest common factor (GCF) is the largest factor that two or more terms have in common. Finding the GCF is a useful technique for simplifying expressions and solving equations.

To find the GCF, list the factors of each term and identify the largest factor that appears in each list. For example, in the denominator of the exercise, the terms are 6x and 2y. The factors of 6 are 1, 2, 3, and 6, and for 2, they are 1 and 2. The GCF is 2. So, the denominator \[6x + 2y\] can be written as \[2(3x + y)\]

• Factor out common factors to simplify expressions.
• Identifying the GCF helps reduce complexity.
• Always check for a GCF before proceeding with other simplification techniques.

sum of cubes

The sum of cubes is a specific algebraic identity used to factorize expressions involving the sum of two cubic terms. The standard formula is: \[a^3 + b^3 = (a + b)(a^2 - ab + b^2)\]

In our example, the expression \[27x^3 + y^3\] was factorized by recognizing that 27x³ is \[(3x)^3\] and y³ is y³. Once identified, apply the formula:

• Identify the cubic terms in the expression.
• Write them as a³ and b³.
• Use the sum of cubes formula to factorize them.

Applying this formula simplifies the process of breaking down complex cubic expressions into simpler binomial and trinomial factors.