Problem 31 - College Algebra with Corequisite Support

Question

For the following exercises, divide the rational expressions. \(\frac{22 y^{2}+59 y+10}{12 y^{2}+28 y-5} \div \frac{11 y^{2}+46 y+8}{24 y^{2}-10 y+1}\)

Step-by-Step Solution

Answer

\(\frac{(2y + 5)(6y - 1)(4y - 1)}{(3y - 1)(4y + 5)(y + 4)}\)

Step 1: Rewrite the Division as Multiplication

We start by rewriting the division of two fractions as multiplication by the reciprocal. This means flipping the second fraction and changing the division sign to a multiplication sign:\[\frac{22y^2 + 59y + 10}{12y^2 + 28y - 5} \times \frac{24y^2 - 10y + 1}{11y^2 + 46y + 8}\]

Step 2: Factor the Numerators and Denominators

Factor each polynomial in the numerators and denominators if possible:- Factor the first numerator: \( (2y + 5)(11y + 2) \).- Factor the first denominator: \( (3y - 1)(4y + 5) \).- Factor the second numerator: \( (6y - 1)(4y - 1) \).- Factor the second denominator: \( (11y + 2)(y + 4) \).

Step 3: Substitute the Factorized Forms

Replace the original expressions with their factorized forms:\[\frac{(2y + 5)(11y + 2)}{(3y - 1)(4y + 5)} \times \frac{(6y - 1)(4y - 1)}{(11y + 2)(y + 4)}\]

Step 4: Cancel Common Factors

Identify and cancel common factors in the numerators and denominators:- \(11y + 2\) is a common factor in both a numerator and a denominator.After canceling \(11y + 2\), the expression simplifies to:\[\frac{(2y + 5)(6y - 1)(4y - 1)}{(3y - 1)(4y + 5)(y + 4)}\]

Step 5: Simplify the Expression

Reassess the remaining expression to see if it can be simplified further. No additional common terms can be canceled between the numerator and the denominator, so this is the most simplified form of the expression.

Key Concepts

Factoring Polynomials

Factoring polynomials is like uncovering the building blocks of an expression. A polynomial can often be expressed as a product of simpler polynomials, known as factors. The aim is to break the given polynomial into parts that multiply to form the original expression. For example, the expression \(22y^2 + 59y + 10\) can be broken down into factors \((2y + 5)(11y + 2)\).

To factor a polynomial, look for the greatest common factor (GCF) among the terms, apply special factoring formulas (like the difference of squares), or use trial and error with the proposed binomials. When factoring quadratics of the form \(ax^2 + bx + c\), consider pairs of factors of \(ac\) whose sum is \(b\).

• Identify the GCF, if there is one, and factor it out.
• For quadratics, find two numbers that multiply to the product \(ac\) and add to \(b\).
• Rewrite the polynomial as a product of binomials.

Polynomial Multiplication

Multiplying polynomials involves distributing each term in the first polynomial by each term in the second polynomial, then combining like terms. Consider the multiplication \((2y + 5)(11y + 2)\). We apply the distributive property:

• First multiply each term in the first polynomial by each term in the second polynomial:

\[2y \times 11y = 22y^2 \2y \times 2 = 4y \5 \times 11y = 55y \5 \times 2 = 10\]Then, combine like terms:

- Combine the \(y\) terms: \(4y + 55y = 59y\).

The resulting expression from multiplying is \(22y^2 + 59y + 10\). This method ensures you account for every term and correctly combine overlapping terms.

• Make use of the FOIL method for binomials: First, Outside, Inside, Last.
• Always combine like terms to simplify where possible.

Division of Polynomials

Dividing one polynomial by another involves rewriting the division as multiplication by the reciprocal of the divisor. This process aligns with how division of fractions or rational expressions is handled. For an expression like \(\frac{22y^2 + 59y + 10}{12y^2 + 28y - 5} \div \frac{11y^2 + 46y + 8}{24 y^2 - 10y + 1}\), transform the division into:
\[\frac{22y^2 + 59y + 10}{12y^2 + 28y - 5} \times \frac{24y^2 - 10y + 1}{11y^2 + 46y + 8}\]This change simplifies the process by turning it into a multiplication problem. Always remember to

• Factorize the polynomials beforehand if possible.
• Multiply by the reciprocal rather than directly dividing.
• Cancel out common factors where applicable.

Simplifying Fractions

Simplifying fractions or rational expressions is all about reducing the expression to its most basic form. This involves factoring both the numerator and the denominator and then canceling out any common factors. For example:
Given:\[\frac{(2y + 5)(11y + 2)}{(3y - 1)(4y + 5)} \times \frac{(6y - 1)(4y - 1)}{(11y + 2)(y + 4)}\]We look for common factors in the numerator and denominator. Here, \((11y + 2)\) appears in both; hence it can be canceled out:
Resulting in:\[\frac{(2y + 5)(6y - 1)(4y - 1)}{(3y - 1)(4y + 5)(y + 4)}\]

This simplicity renders the fraction easier to understand and work with. Always:

• Factorize completely before canceling.
• Simplify step-by-step to avoid mistakes.
• Verify the final expression to ensure it is reduced to the simplest form.