Problem 16 - Algebra A Combined Function

Question

Find each product and simplify if possible. See Examples 1 through 3. $$ \frac{x^{2}+9 x+20}{x^{2}-15 x+44} \cdot \frac{x^{2}-11 x+28}{x^{2}+12 x+35} $$

Step-by-Step Solution

Answer

\(\frac{(x+4)(x-7)}{(x-11)(x+7)}\) is the simplified product.

Step 1: Factor the Numerator of the First Fraction

Look at the numerator of the first fraction, which is \(x^2+9x+20\). To factor it, find two numbers that multiply to 20 and add to 9. These numbers are 4 and 5, so we can factor it as \((x+4)(x+5)\).

Step 2: Factor the Denominator of the First Fraction

Now, factor the denominator \(x^2-15x+44\). Find two numbers that multiply to 44 and add to -15. The numbers -4 and -11 satisfy these conditions, so the denominator factors as \((x-4)(x-11)\).

Step 3: Factor the Numerator of the Second Fraction

Consider the numerator of the second fraction \(x^2-11x+28\). You need numbers that multiply to 28 and add to -11, which are -4 and -7. This factors as \((x-4)(x-7)\).

Step 4: Factor the Denominator of the Second Fraction

Look at \(x^2+12x+35\). Find numbers that multiply to 35 and add to 12. These are 5 and 7, so it factors to \((x+5)(x+7)\).

Step 5: Multiply the Fractions

Rewrite the fractions with their factors: \[ \frac{(x+4)(x+5)}{(x-4)(x-11)} \cdot \frac{(x-4)(x-7)}{(x+5)(x+7)} \] Now multiply across the numerators and denominators: \[ \frac{(x+4)(x+5)(x-4)(x-7)}{(x-4)(x-11)(x+5)(x+7)} \]

Step 6: Simplify the Expression by Canceling Common Factors

Identify and cancel the common factors in the numerator and the denominator. The factors \((x-4)\) and \((x+5)\) appear in both, so they can be canceled out. This leaves: \[ \frac{(x+4)(x-7)}{(x-11)(x+7)} \]

Step 7: Write the Simplified Result

After canceling common factors, the simplified product of the original fractions is \(\frac{(x+4)(x-7)}{(x-11)(x+7)}\).

Key Concepts

Factoring Trinomials

Factoring trinomials is an essential skill in algebra, especially when working with rational expressions. Basically, a trinomial is composed of three terms, typically in the form of \( ax^2 + bx + c \). The goal is to rewrite this expression as a product of two binomials. For example, you start with \( x^2 + 9x + 20 \). To factor this, look for two numbers that multiply to 20 (the constant term) and add up to 9 (the linear coefficient). These numbers are 4 and 5, so the trinomial factors to \((x+4)(x+5)\).

Similarly, for \( x^2 - 15x + 44 \), find two numbers that multiply to 44 and add to -15; the right numbers are -4 and -11, leading to the factorization \((x-4)(x-11)\). By breaking down polynomials like this, you make the multiplication and simplification processes much easier.

Multiplying Fractions

Multiplying fractions involves multiplying the numerators together to make a new numerator and the denominators together to make a new denominator. With algebraic fractions, this principle remains the same, but with variables involved. Once each expression is broken into its factors, like \( \frac{(x+4)(x+5)}{(x-4)(x-11)} \text{ and } \frac{(x-4)(x-7)}{(x+5)(x+7)} \), you multiply across the numerators to get \((x+4)(x+5)(x-4)(x-7)\), and across the denominators to get \((x-4)(x-11)(x+5)(x+7)\).

This gives you a new single fraction rather than keeping two separate fractions, each set of terms multiplied out.

Simplifying Expressions

Simplifying expressions is about reducing them to their most condensed form without altering their value. Once you have multiplied the fractions, you often end up with terms that can cancel each other out.

In this example, after factoring, multiplying led to \( \frac{(x+4)(x+5)(x-4)(x-7)}{(x-4)(x-11)(x+5)(x+7)} \). Notice here, \((x-4)\) appears in both the numerator and the denominator, as does \((x+5)\). These common factors can be canceled, leaving you with \( \frac{(x+4)(x-7)}{(x-11)(x+7)} \). Canceling common terms simplifies the expression and makes further calculations easier and less prone to error.

Algebraic Fractions

Algebraic fractions are similar to numerical fractions; they express a quotient of polynomials. Working with them requires a good understanding of polynomial operations like addition, subtraction, multiplication, and division. Fractions like \( \frac{x^2+9x+20}{x^2-15x+44} \) involve factoring, identifying restrictions where the fractions become undefined (for instance, when the denominator equals zero), and simplifying for easier calculations.

In our example, it became necessary to factor each polynomial, multiply them, and then simplify by canceling common factors. This made the expression manageable without changing its original meaning. Thus, manipulating algebraic fractions often involves these key processes to express the rational expression in its simplest form.