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Chapter 6: Problem 50

Write each rational expression in lowest terms. $$ \frac{16 x^{2}-9 y^{2}}{12 x-9 y} $$

Short Answer

Expert verified
\[ \frac{4x + 3y}{3} \]

Step by step solution

01

Factor numerator

The numerator is a difference of squares. Factor it using the formula \[a^2 - b^2 = (a - b)(a + b)\]. Here, \[16x^2 - 9y^2\] becomes \[(4x - 3y)(4x + 3y)\].
02

Simplify the denominator

Check if the denominator can be factored. The denominator \[12x - 9y\] can be factored by pulling out the greatest common factor (GCF). The GCF of 12 and 9 is 3, so \[12x - 9y = 3(4x - 3y)\].
03

Cancel common factors

Now simplify the rational expression by canceling the common factors in the numerator and the denominator. The expression is now \[ \frac{(4x - 3y)(4x + 3y)}{3(4x - 3y)}\]. Cancel \[4x - 3y\], resulting in \[ \frac{4x + 3y}{3} \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

factoring polynomials
Factoring polynomials is a key skill in algebra. It involves breaking down a polynomial into simpler components known as factors. These factors, when multiplied together, give you the original polynomial. This process helps in simplifying rational expressions and solving equations.
To factor a polynomial:
  • First, look for the greatest common factor (GCF) among the terms.
  • Then, use different factoring techniques such as grouping, using formulas like the difference of squares, or factoring trinomials.
In our example, we factored the numerator \[16x^2 - 9y^2 \] as \[ (4x - 3y)(4x + 3y) \]. This is done using the difference of squares formula.
greatest common factor
The greatest common factor (GCF) is the largest factor that two or more numbers have in common. Finding the GCF is crucial when simplifying expressions, as it helps reduce them to their simplest form.
To find the GCF between two numbers:
  • List the factors of each number.
  • Identify the largest factor that appears in both lists.
In our example, the denominator \[12x - 9y \] has a GCF of 3. We factor it out, turning \[12x - 9y \] into \[3(4x - 3y) \].
difference of squares
The difference of squares is a specific way to factor expressions that fit the form \[a^2 - b^2 \].
The formula for factoring a difference of squares is: \[a^2 - b^2 = (a - b)(a + b) \].
This formula is useful because it turns a single complicated expression into simpler, multiplied terms.
For instance, in our problem, the numerator \[16x^2 - 9y^2 \] is a difference of squares. We used the formula to factor it as \[ (4x - 3y)(4x + 3y) \]. This step allows us to further simplify the rational expression by canceling out common factors.

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Most popular questions from this chapter

Write four equivalent forms for each rational expression. $$ -\frac{5 x-6}{x+4} $$

Multiply or divide. Write each answer in lowest terms. $$ \frac{m^{2}+3 m+2}{m^{2}+5 m+4} \cdot \frac{m^{2}+10 m+24}{m^{2}+5 m+6} $$

Multiply. Write each answer in lowest terms. $$ \frac{(x-y)^{2}}{2} \cdot \frac{24}{3(x-y)} $$

Add or subtract as indicated. Write each answer in lowest terms. $$ \frac{1}{2}+\frac{7}{8} $$

Multiply. Write each answer in lowest terms. $$ \frac{(a+b)^{2}}{5} \cdot \frac{30}{2(a+b)} $$
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