Problem 83 - Intermediate Algebra

Question

Perform the operations and simplify. $$ \left(4 x^{2}-9\right) \div \frac{2 x^{2}+5 x+3}{x+2} \cdot \frac{1}{2 x-3} $$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{x+2}{x+1}\).

Step 1: Understand the Given Expression

The given expression involves division and multiplication of rational expressions: \[\left(4x^2 - 9\right) \div \frac{2x^2 + 5x + 3}{x+2} \cdot \frac{1}{2x-3}\] This requires rewriting as a multiplication for simplification.

Step 2: Rewrite Division as Multiplication

To simplify, first rewrite the division as a multiplication: \[\left(4x^2 - 9\right) \cdot \frac{x+2}{2x^2 + 5x + 3} \cdot \frac{1}{2x-3}\] Now, we can simplify by factoring the numerator and denominators where possible.

Step 3: Factor Polynomials

Factorize each polynomial:- \(4x^2 - 9\) is a difference of squares, thus \((2x+3)(2x-3)\).- \(2x^2 + 5x + 3\) factors to \((2x+3)(x+1)\).Substituting these factorizations into the expression gives:\[\left((2x+3)(2x-3)\right) \cdot \frac{x+2}{(2x+3)(x+1)} \cdot \frac{1}{2x-3}\]

Step 4: Simplify the Expression

Cancel out common factors in the numerator and the denominator:- Cancel \(2x+3\) in the numerator of the first part and the denominator of the second part.- Cancel \(2x-3\) in the numerator of the first part and the denominator of the third part.The expression simplifies to:\[\frac{x+2}{x+1}\]

Step 5: Final Simplification Check

Verify no further simplifications can be made or common factors canceled. Since there are no repeating factors in the remaining fraction, the expression is fully simplified.

Key Concepts

Factoring Polynomials

Factoring polynomials is like breaking down a number into its prime factors, only with variables and their coefficients. A polynomial is an algebraic expression consisting of terms made up of variables raised to a power, with coefficients. When you factor a polynomial, you rewrite it as a product of simpler expressions, called factors. This is particularly useful when simplifying expressions.
To factor polynomials:

• First, look for a greatest common factor (GCF) in all the terms. If one exists, factor it out.
• Identify special patterns such as the difference of squares or perfect square trinomials. For example, 4x² - 9 is a difference of squares because it can be rewritten as (2x)² - (3)², which factors into (2x+3)(2x-3).
• If the polynomial is a trinomial (with 3 terms), like 2x² + 5x + 3, find two binomials that multiply to make the original trinomial. These often involve looking for two numbers that multiply to the constant term (3) and add to the linear coefficient (5) as is done with quadratic trinomials.

Practicing these techniques makes factoring faster and more intuitive over time.

Simplifying Rational Expressions

Simplifying rational expressions is much like simplifying fractions. It involves rewriting the expression into its simplest form by canceling out any common factors in the numerator and denominator. Rational expressions are fractions where the numerator and/or the denominator are polynomials.
To simplify:

• Factorize both the numerator and the denominator completely.
• Identify and cancel out all common factors shared by the numerator and the denominator. This reduces the fraction to its simplest form.
• Be careful not to cancel terms that aren't factors. Only terms multiplied together can be canceled.

For example, after factoring the expression in our original exercise, the terms (2x+3) and (2x-3) canceled out from both the numerator and denominator, which simplified the expression significantly. This step is crucial to achieve the simplest expression.

Division and Multiplication of Polynomials

When working with polynomial expressions, division and multiplication often appear together, especially in complex fractions or rational expressions.
For division:

• Dividing by a fraction is the same as multiplying by its reciprocal. This is the first step when simplifying expressions involving division of polynomials.
• Switch the division operation into multiplication by flipping the second fraction.

For multiplication:

• Multiply the numerators together and the denominators together separately.
• Simplify the new expression by factoring and canceling common terms if possible.

In the provided exercise, division is rewritten as multiplication by taking the reciprocal of the second fraction. Once in a multiplication-only format, the problem can be tackled through factoring and then simplifying. Understanding this process is key to mastering operations with polynomials.