Problem 46 - Introductory Algebra for College Students

Question

Divide as indicated. $$\left(y^{2}+4 y-5\right)+\frac{y^{2}-25}{y+7}$$

Step-by-Step Solution

Answer

The solution to the division problem is \(\frac{y^{3}+11 y^{2} + 23y -35}{y^{2} -25}\)

Step 1: Simplify the numerator and denominator

Before starting the division, we need to simplify each of the numerator and the denominator. The numerator can't be factorised, so it remains as \(y^{2}+4 y-5\). The denominator can be factored as \((y-5)(y+5)/(y+7)\) using differences of squares.

Step 2: Division

Now, divide the numerator by the denominator. We express this division as multiplication by the reciprocal of the denominator. This gives us the expression \((y^{2}+4 y-5)(y+7)/(y-5)(y+5)\)

Step 3: Simplify the Expression

In this step, we expand and simplify the expression as far as possible. This gives \((y^{3}+7 y^{2} +4 y^{2} + 28y -5y -35)/(y^2 -25)\), which simplifies further to \(y^{3}+11 y^{2} + 23y -35)/(y^2 -25)\)

Key Concepts

Polynomial Division

When faced with the problem of dividing one polynomial by another, you essentially apply similar techniques to those used in long division with numbers. However, instead of numbers, you work with polynomials, which are mathematical expressions consisting of variables and coefficients.
In polynomial division, you're often required to divide a large polynomial by a smaller one to obtain a quotient and sometimes a remainder. To do this,

• Write the division as a fraction.
• Simplify if necessary by factoring or simplifying the numerator and the denominator.
• Perform the division step by step, akin to long division, by dividing the terms individually.

In the original exercise, we start by analyzing the structure of the fractions given. You can simplify the division process by turning it into a multiplication using the reciprocal of the divisor. This simplifies calculations and helps identify factors and potential simplifications.

Factoring Polynomials

Factoring polynomials is a method used to break down expressions into more manageable pieces. It's similar to finding the prime factors of a number. Let's take a closer look:

• Identify simple forms such as difference of squares, where you have expressions like \[a^2 - b^2 = (a + b)(a - b).\]
• For trinomials, check if they can be decomposed into the product of binomials.
• Look for a common factor in each term.

In our case, we attempted to factor parts of our given equation. The expression \(y^2 - 25\) can be expressed as \((y - 5)(y + 5)\) via the difference of squares method. This crucial step not only simplified the process but allowed us to organize our equation effectively for further simplification.

Simplifying Expressions

Once you've managed to express and factor your polynomials correctly, the next step is simplifying those expressions, aiming for the most reduced form possible.

• Expand any numerators and denominators if necessary and simplify each term.
• Cancel out any common terms in the numerator and denominator.
• Rearrange or combine like terms to achieve the simplest form.

After factoring the denominator, our expression was rewritten using the reciprocal of the divisor, which led to a multiplication rather than complex division. Through expanding and combining like terms, the polynomials were further simplified to \(y^3 + 11y^2 + 23y - 35\). Simplifying helps us achieve a clearer, more concise representation of the original problem, thus providing a deeper understanding of the relationships between the terms.