Problem 22 - College Algebra with Corequisite Support

Question

For the following exercises, multiply the rational expressions and express the product in simplest form. \(\frac{2 n^{2}-n-15}{6 n^{2}+13 n-5} \cdot \frac{12 n^{2}-13 n+3}{4 n^{2}-15 n+9}\)

Step-by-Step Solution

Answer

The simplified form is \(\frac{(2n + 5)(3n - 1)}{(3n + 5)(2n - 1)}\).

Step 1: Factor the Numerators and Denominators

Start by factoring each numerator and denominator separately. For \(2n^2 - n - 15\), find two numbers that multiply to \(-30\) (the product of \(2\) and \(-15\)) and add to \(-1\). These numbers are \(-6\) and \(5\). Rewrite and factor the expression:\[2n^2 - n - 15 = (2n + 5)(n - 3)\]For \(12n^2 - 13n + 3\), find two numbers that multiply to \(36\) (the product of \(12\) and \(3\)) and add to \(-13\). These numbers are \(-9\) and \(-4\). Rewrite and factor the expression:\[12n^2 - 13n + 3 = (3n - 1)(4n - 3)\]Now factor the denominators:For \(6n^2 + 13n - 5\), find two numbers that multiply to \(-30\) and add to \(13\). These numbers are \(15\) and \(-2\). Rewrite and factor the expression:\[6n^2 + 13n - 5 = (3n + 5)(2n - 1)\]For \(4n^2 - 15n + 9\), find two numbers that multiply to \(36\) (the product of \(4\) and \(9\)) and add to \(-15\). These numbers are \(-12\) and \(-3\). Rewrite and factor the expression:\[4n^2 - 15n + 9 = (4n - 3)(n - 3)\]

Step 2: Write Down the Factored Expression

Replace each part of the original expression with its factored form from Step 1:\[\frac{(2n + 5)(n - 3)}{(3n + 5)(2n - 1)} \cdot \frac{(3n - 1)(4n - 3)}{(4n - 3)(n - 3)}\]

Step 3: Cancel Common Factors

Identify and cancel the common factors in the numerators and denominators:- Cancel \(n - 3\) from the first numerator and the second denominator.- Cancel \(4n - 3\) in the second fraction.This simplifies to:\[\frac{(2n + 5)}{(3n + 5)(2n - 1)} \cdot \frac{(3n - 1)}{1}\]

Step 4: Simplify the Remaining Expression

Multiply the remaining numerators and denominators together:Numerator:\[(2n + 5)(3n - 1)\]Denominator:\[(3n + 5)(2n - 1)\]So, the expression simplifies to:\[\frac{(2n + 5)(3n - 1)}{(3n + 5)(2n - 1)}\]

Step 5: Final Check for Further Simplification

Double-check if any further simplification can be done. Since there are no common factors between the remaining numerator and denominator, the expression is in its simplest form.

Key Concepts

Factoring Polynomials

Factoring polynomials is a crucial step when dealing with rational expressions because it helps to simplify the expressions by revealing their common factors. In simple terms, factoring is the process of breaking down a complex expression into simpler components, called factors, which multiply together to give the original polynomial.

For example, let's consider the polynomial \(2n^2 - n - 15\). To factor this expression, we need to find two numbers that both multiply to the product of the leading coefficient and the constant term, which is \(-30\), and add to the linear coefficient, \(-1\). These numbers are \(-6\) and \(5\). By rewriting the middle term, we transform the expression into \((2n + 5)(n - 3)\).

• Always find numbers that multiply to the product of the first and last terms' coefficients.
• Aim for these numbers to add up to the middle term's coefficient.
• Rewrite and group the terms to facilitate factoring.

Through consistent practice, factoring becomes much easier, leading to swift and accurate simplification of rational expressions.

Multiplying Fractions

Once the polynomials are factored, the next step is understanding how to multiply fractions, which in the context of rational expressions, involves the multiplication of their numerators and denominators. The general rule is simple: multiply the numerators together, and then do the same for the denominators.

For instance, consider the fractions in the expression \(\frac{(2n + 5)(n - 3)}{(3n + 5)(2n - 1)} \cdot \frac{(3n - 1)(4n - 3)}{(4n - 3)(n - 3)}\). To find their product, we initially think about multiplying the tops (numerators) together and then the bottoms (denominators) together.

• Multiply numerators across to get the new numerator.
• Multiply denominators in the same manner to find the new denominator.

This step might seem straightforward but pay attention to the opportunity to cancel out common factors.
Use this operation to also simplify fractions before doing any unnecessary computations.

Simplifying Expressions

Simplifying expressions in rational algebra involves reducing fractions to their lowest terms by removing any common factors in the numerators and denominators. This keeps calculations simpler and results cleaner.

Let's dive into the expression \(\frac{(2n + 5)(3n - 1)}{(3n + 5)(2n - 1)}\) produced after factoring and cancelling common terms. Once common factors like \((n - 3)\) and \((4n - 3)\) are canceled out, we arrive here. To check for more simplification, verify there are no further common divisors.

• Firstly, identify and cancel out common factors from the numerator and the denominator.
• After removing common elements, ensure whether the remaining parts still have factors that can be simplified.
• The goal is to retain the simplest form possible.

Comprehending these steps fosters deeper insight into algebraic manipulation and helps prevent errors during calculations, ultimately aiding clearer and correct outcomes.