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Chapter 7: Problem 88

Simplify. $$\frac{2 a^{2}-5 a b}{c-3 d} \div\left(4 a^{2}-25 b^{2}\right)$$

Short Answer

Expert verified
\[ \frac{2a^2 - 5ab}{(c - 3d)(2a - 5b)(2a + 5b)} \]

Step by step solution

01

Rewrite the Division as Multiplication

\[ \frac{2a^2 - 5ab}{c - 3d} \times \frac{1}{4a^2 - 25b^2} \]
02

Factor the Denominator of the Second Fraction

Recognize that \( 4a^2 - 25b^2 \) is a difference of squares. Therefore, it factors to: \[ 4a^2 - 25b^2 = (2a - 5b)(2a + 5b) \] So the expression becomes: \[ \frac{2a^2 - 5ab}{c - 3d} \times \frac{1}{(2a - 5b)(2a + 5b)} \]
03

Simplify

Observe that there are no common factors between the numerator and the denominator across the two fractions that can be canceled out. Hence, the simplified form of the expression is: \[ \frac{2a^2 - 5ab}{(c - 3d)(2a - 5b)(2a + 5b)} \]

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Key Concepts

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

Simplifying Fractions
Simplifying fractions means making the fraction easier to understand and use by reducing it to its simplest form. In the context of rational expressions, simplifying often involves:
  • Finding common factors in the numerator and denominator.
  • Canceling out these common factors.
For example, if you have the fraction: \[ \frac{10}{25} \] You can simplify it by dividing both the numerator and the denominator by their greatest common divisor, which is 5. This gives: \[ \frac{10 \text{ 梅 } 5}{25 \text{ 梅 } 5} = \frac{2}{5} \] It's the same concept applied to rational expressions, except with polynomials or variables. It's crucial to identify and cancel out common factors to simplify the expression effectively.
Polynomial Division
Polynomial division is a process similar to long division but with polynomials. It involves:
  • Dividing the terms of the polynomials step by step.
  • Subtracting the results to find the remainder.
In the given exercise, we convert division into multiplication by flipping the second fraction. This is because dividing by a fraction is the same as multiplying by its reciprocal (the inverse). So, if we have: \[ \frac{2a^2 - 5ab}{c - 3d} \text{ 梅 } (4a^2 - 25b^2) \] We rewrite it as: \[ \frac{2a^2 - 5ab}{c - 3d} \times \frac{1}{(4a^2 - 25b^2)} \] This step transforms division into a simpler multiplication process, which makes the next steps more manageable.
Factoring
Factoring is a technique used to break down expressions into simpler, multiplied components. This is crucial when dealing with polynomial expressions. Some common factoring methods include:
  • Finding common factors: For example, \[ 2a^2 - 4a = 2a(a - 2) \]
  • Factoring by grouping: Group terms to find common factors in parts of the expression.
  • Difference of squares: This applies when you have an expression like \[ A^2 - B^2 \], which factors to \[ (A + B)(A - B) \].
In our exercise, we factor \[ 4a^2 - 25b^2 \] as a difference of squares. Therefore: \[ 4a^2 - 25b^2 = (2a - 5b)(2a + 5b) \] This factoring step is essential to simplifying the rational expression further, as it reveals the underlying factors clearly, enabling easier simplification.

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