Problem 18 - Intermediate Algebra

Question

Fill in the blanks: $$\frac{-2 x^{2}-4 x}{(x+7)(x-9)}=\frac{\square(x+2)}{(x+7)(x-9)}=\square\frac{2 x(x+2)}{(x+7)(x-9)}$$

Step-by-Step Solution

Answer

Fill in: -2x; -1.

Step 1: Factor the Numerator

The expression given is \(-2x^2 - 4x\). We need to factor this quadratic expression. Start by taking \(-2x\) common from both terms. This gives, \(-2x(x + 2)\).

Step 2: Rewrite the Expression

Now that we have factored \(-2x^2 - 4x\) into \(-2x(x + 2)\), we can substitute this in the numerator of the fraction: \[\frac{-2x(x+2)}{(x+7)(x-9)}\].

Step 3: Identify Missing Factors

Next, compare the expression \(\frac{-2x(x+2)}{(x+7)(x-9)}\) to \(\frac{\square(x+2)}{(x+7)(x-9)}\) and \(\square\frac{2x(x+2)}{(x+7)(x-9)}\). In both cases, we need to determine how to express the factor we found (-2x) as blank spaces.

Step 4: Fill in the Blanks

For \(\frac{\square(x+2)}{(x+7)(x-9)}\), since we have \(-2x(x+2)\), the missing factor is \(-2x\). For \(\square\frac{2x(x+2)}{(x+7)(x-9)}\), since dividing by 2x gives \(-1\), the missing value is \(-1\).

Key Concepts

Polynomials

Polynomials are an essential part of algebra and mathematics in general. A polynomial is a mathematical expression made up of variables, coefficients, and exponents combined using addition, subtraction, and multiplication. Each unique combination of these elements is known as a term, and a polynomial can contain one or more terms. For example, the polynomial

• \(-2x^2 - 4x\)

contains two terms:

• \(-2x^2\)
• \(-4x\)

To factor such a polynomial, identify common factors in its terms. You can write it as a product of its factors, for instance:

• \(-2x(x + 2)\)

This polynomial translates into a quadratic expression where

• \(x\)
• \(x^2\)

represent variables raised to different powers. Factoring helps break down polynomials, making them easier to manipulate or solve, especially when simplifying equations and fractions.

Mathematical Expressions

A mathematical expression is a combination of numbers, variables, operators (such as +, -, *, /), and sometimes grouping symbols (like parentheses) that represents a specific value. Expressions are the building blocks of equations. They detail how numbers and symbols are related to one another. When solving problems involving expressions, the goal is often to simplify them or to find values that satisfy them under certain conditions!In the problem provided, you started with an expression:

• \(-2x^2 - 4x\)

Simplifying this, you factor it into:

• \(-2x(x + 2)\)

Expressing or rewriting mathematical expressions often helps in identifying patterns or simplifying problems, especially when simplifying fractions, factoring or solving equations. Working with expressions involves parentheses often to denote specific operations; in this case, grouping was useful for factoring purposes.

Fractions

Fractions are a way to represent numbers that are not whole numbers. They consist of two parts: the numerator and the denominator. In general, a fraction is written as \(\frac{a}{b}\), where \(a\) is the numerator and \(b\) is the denominator. Fractions can represent parts of a whole, ratios, or division of numbers. When dealing with algebraic fractions, simplifying by factoring both the numerator and denominator can make calculations easier.In our equation, the fraction is:

• \(\frac{-2x(x+2)}{(x+7)(x-9)}\)

Through factoring, we turned a complex numerator

• \(-2x^2 - 4x\)

into its simpler form, making it easier to substitute back into the fraction. When working with fractions, especially in equations, finding common denominators and factoring the numerators are key steps to simplifying and comparing the expressions efficiently. By understanding how fractions can be broken down and manipulated, you improve your ability to solve equations that include fractional forms.