Problem 55 - Introductory Algebra for College Students

Question

Divide as indicated. $$\frac{y^{2}+5 y+4}{y^{2}+12 y+32}+\frac{y^{2}-12 y+35}{y^{2}+3 y-40}$$

Step-by-Step Solution

Answer

The simplified form is \(\frac{2y-6}{y+8}.\)

Step 1: Factorize the polynomials

Factor each of the four quadratic polynomials: \(y^{2}+5 y+4=(y+4)(y+1)\), \(y^{2}+12 y+32=(y+8)(y+4)\), \(y^{2}-12 y+35=(y-5)(y-7)\), \(y^{2}+3 y-40=(y+8)(y-5)\)

Step 2: Rewrite the original expression with the factorized polynomials

Now replace the original polynomials in the expression with their factorized versions, the expression becomes: \(\frac{(y+4)(y+1)}{(y+8)(y+4)} + \frac{(y-5)(y-7)}{(y+8)(y-5)}\)

Step 3: Simplify the fractions

Notice that the numerator of the first fraction has a common factor with the denominator of the first fraction, and the numerator of the second fraction has a common factor with the denominator of the second fraction. These can be cancelled: \(\frac{y+1}{y+8} + \frac{y-7}{y+8} \)

Step 4: Combine the fractions

Now both fractions have the same denominator, so they can be combined into a single fraction: \(\frac{(y+1)+(y-7)}{y+8}\)

Step 5: Simplify the numerator

Simplify the numerator by combining like terms, this results in: \(\frac{2y-6}{y+8}\)

Key Concepts

Polynomial Factorization

Polynomial factorization is an essential step in solving problems involving algebraic fractions. It involves breaking down a polynomial into a product of simpler polynomials, known as factors. This process can simplify complex expressions and make them easier to handle.

For example, in the expression \( y^2 + 5y + 4 \), we used factorization to express it as \( (y + 4)(y + 1) \). This was accomplished by finding two numbers that multiply to the constant term (4) and add up to the linear coefficient (5).

The ability to factor polynomials quickly is crucial for simplifying fractions and solving equations efficiently. Remember, always look for the greatest common factor (GCF) first before breaking down more complex polynomials.

Simplifying Fractions

Simplifying fractions comes down to canceling common factors from the numerator and denominator. Once polynomials are factorized, simplifying is often the next step.

In our exercise, we factored the first fraction as \( \frac{(y+4)(y+1)}{(y+8)(y+4)} \). Here, the \( y+4 \) term was present in both the numerator and denominator, allowing us to cancel it out, leaving \( \frac{y+1}{y+8} \).

Always make sure the factor you're canceling is not zero, as division by zero is undefined. Simplifying not only makes the expression easier to work with but also often reveals easier methods to solve the rest of the problem.

Combining Like Terms

Combining like terms refers to the process of summing up terms in an expression that have the same variable parts. This usually comes into play after simplifying fractions or when trying to make expressions more concise.

In the final steps of the problem, after simplifying the fractions, we combined the terms \( y+1 \) and \( y-7 \) to form \( 2y - 6 \).

To correctly combine like terms, remember to consider both the coefficients and the variables. Doing so ensures you have a simplified and correct expression, which is vital for accurately solving algebraic equations.

Quadratic Polynomials

Quadratic polynomials are polynomials of degree two and have the general form \( ax^2 + bx + c \). Solving problems involving quadratic polynomials often requires understanding their properties and methods for factorization.

Each quadratic polynomial in this exercise was expressed as a product of two binomial factors. Recognizing patterns, such as whether the quadratic can be factored using techniques like the difference of squares or perfect square trinomials, greatly assists in the factorization process.

Mastering quadratic polynomials and their factorization techniques provides a strong foundation for solving a wide range of algebraic problems. It's a fundamental skill that connects to many other algebraic concepts.