Problem 29 - Algebra A Combined Function

Question

Perform each indicated operation. Simplify if possible. \(\frac{3 x^{4}}{7}-\frac{4 x^{2}}{21}\)

Step-by-Step Solution

Answer

The simplified form is \(\frac{x^2(3x - 2)(3x + 2)}{21}\).

Step 1: Identify a Common Denominator

To perform the subtraction, we first identify a common denominator for the fractions. The denominators are 7 and 21. The least common multiple of 7 and 21 is 21, so we will use 21 as the common denominator.

Step 2: Rewrite Each Fraction

Rewrite each fraction with 21 as the denominator. The first fraction becomes \(\frac{3x^4 \cdot 3}{21} = \frac{9x^4}{21}\). The second fraction \(\frac{4x^2}{21}\) remains the same because it already has the common denominator.

Step 3: Subtract the Fractions

Subtract the second fraction from the first: \(\frac{9x^4}{21} - \frac{4x^2}{21}\). Since the denominators are the same, keep the denominator and subtract the numerators: \(\frac{9x^4 - 4x^2}{21}\).

Step 4: Simplify the Expression

Factor the numerator. \(9x^4 - 4x^2 = x^2(9x^2 - 4)\). Notice that \(9x^2 - 4\) is a difference of squares: \((3x)^2 - 2^2\), which factors to \((3x - 2)(3x + 2)\). Thus, the expression simplifies to \(\frac{x^2(3x - 2)(3x + 2)}{21}\).

Step 5: Check for Further Simplification

Check if any further cancellation can be made. In this case, there are no common factors between the numerator and the denominator, so this is the simplest form.

Key Concepts

Common Denominator

When dealing with fractions, a common denominator is essential to perform addition or subtraction. The common denominator is a shared multiple of the denominators in each fraction. This allows you to compare or combine the fractions easily. For instance, with the fractions \( \frac{3x^4}{7} \) and \( \frac{4x^2}{21} \), the denominators are 7 and 21.
The least common multiple (LCM) of these numbers must be found. The LCM of 7 and 21 is 21, because 21 is the smallest number that both 7 and 21 divide evenly into.

• Find the least common multiple of the denominators.
• Use this LCM as the new common denominator.

The original denominators are transformed by multiplying them and the numerator accordingly, preserving the value of the fractions and permitting operations between them.

Fraction Subtraction

Subtracting fractions involves aligning them with a common denominator and then directly subtracting the numerators. For example,

1. Start by asserting a common denominator. For \( \frac{3x^4}{7} \) and \( \frac{4x^2}{21} \), the common denominator is 21.
2. Transform the fractions: the first one becomes \( \frac{9x^4}{21} \) as you multiply both numerator and denominator by 3.

The operation then becomes straightforward:

• Keep the denominator (21).
• Subtract the numerators: \( 9x^4 - 4x^2 \).

Thus, the resulting fraction is \( \frac{9x^4 - 4x^2}{21} \). Ensuring the same denominator simplifies subtraction to a basic arithmetic operation on the numerators.

Factorization

Factorization simplifies mathematical expressions by breaking them into factors. Consider the expression \( 9x^4 - 4x^2 \) from the previous fraction subtraction.
First, observe common factors in the terms: both terms contain \( x^2 \).

• Factor out \( x^2 \), resulting in \( x^2 (9x^2 - 4) \).
• Notice that \( 9x^2 - 4 \) is a difference of squares.

The difference of squares can be factored further using the formula: \[ a^2 - b^2 = (a-b)(a+b) \] In our problem:

• \( 9x^2 = (3x)^2 \) and \( 4 = 2^2 \).
• Apply the formula: \( 9x^2 - 4 = (3x - 2)(3x + 2) \).

Finally, the entire expression is factored as \( \frac{x^2 (3x - 2)(3x + 2)}{21} \). This reduced form can often be easier to work with in further calculations.