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Magazine Chapter 11: Problem 52
In Exercises \(5-52,\) express each of the given expressions in simplest form with only positive exponents. $$4(2 x-1)(x+2)^{-1}-(2 x-1)^{2}(x+2)^{-2}$$
Short Answer
Expert verified
\( \frac{4x^2 + 16x - 9}{(x+2)^2} \) with positive exponents.
Step by step solution
01
Expand Each Term
Start by expanding the terms in the expression separately. The expression is given as \[ 4(2x-1)(x+2)^{-1} - (2x-1)^2(x+2)^{-2}. \]The first term, \(4(2x-1)(x+2)^{-1}\), involves a product of \(4, (2x-1)\), and \((x+2)^{-1}\). Since \((x+2)^{-1}\) is the same as dividing by \((x+2)\), the expression remains the same for further simplification.The second term is \((2x-1)^2(x+2)^{-2}\). Expanding \((2x-1)^2\) using the formula \((a-b)^2 = a^2 - 2ab + b^2\), we have:\[ (2x-1)^2 = 4x^2 - 4x + 1. \]The expression becomes \((4x^2 - 4x + 1)(x+2)^{-2}\).
02
Simplify the Expression
Combine the expanded terms into a single expression: \[ \frac{4(2x-1)}{x+2} - \frac{4x^2 - 4x + 1}{(x+2)^2}. \]To simplify, we need a common denominator, which is \((x+2)^2\).
03
Common Denominator and Combine Terms
Write \(\frac{4(2x-1)}{x+2}\) with the common denominator \((x+2)^2\). Multiply the numerator and denominator by \(x+2\) to obtain:\[ \frac{4(2x-1)(x+2)}{(x+2)^2}. \]Now, rewrite the expression as:\[ \frac{4(2x-1)(x+2) - (4x^2 - 4x + 1)}{(x+2)^2}. \]
04
Expand and Simplify the Numerator
Expand \(4(2x-1)(x+2)\).First, expand \[ (2x-1)(x+2) = 2x^2 + 4x - x - 2 = 2x^2 + 3x - 2. \]Then multiply by 4: \[ 4(2x^2 + 3x - 2) = 8x^2 + 12x - 8. \]Simplify the expression:\[ 8x^2 + 12x - 8 - (4x^2 - 4x + 1). \]Simplify the expression to:\[ 8x^2 + 12x - 8 - 4x^2 + 4x - 1 = 4x^2 + 16x - 9. \]
05
Write in Simplest Form
The final simplified expression, with positive exponents, remains:\[ \frac{4x^2 + 16x - 9}{(x+2)^2}. \]This is the simplest form of the given expression, using only positive exponents.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Simplifying Expressions
When simplifying algebraic expressions, your main goal is to transform a complex expression into its simplest form. This involves combining like terms, reducing fractions if possible, and using basic operation rules.
Here are some steps to consider:
Here are some steps to consider:
- Identify Like Terms: These are terms within an expression that have the same variable raised to the same power. Combine them to simplify the expression further.
- Use Distributor Properties: In this exercise, you first expand products involving coefficient terms and binomials. Think of it like spreading the single term over all parts of another term or expression.
- Simplify Fractions: Express complicated fractions or divisions into a simpler form by removing common factors or by finding common denominators, as will be discussed later.
Exponents
Exponents represent repeated multiplication and are a key part of many algebraic operations. They can turn the process of multiplication into a compact form and help in efficiently handling large numbers.
In the problem you're tackling:
In the problem you're tackling:
- Positive Exponents: Express numbers and variables using only positive exponents. For instance, substituting negative exponents with their reciprocal form helps manage expressions in a simpler, more standardized form.Examples: \[ a^{-n} = \frac{1}{a^n} \]
- Simplification with Exponents: Pay attention to the laws of exponents. When exponents are used in multiplication, the powers are added, and when in division, they are subtracted.Examples:\[ x^{m} \cdot x^{n} = x^{m+n}, \quad \frac{x^m}{x^n} = x^{m-n} \]
Polynomial Operations
Dealing with polynomials often involves operations like addition, subtraction, multiplication, and sometimes division of expressions. A polynomial is essentially an expression consisting of variables, constants, and exponents arranged as a sum of terms.
In this exercise, you primarily perform:
In this exercise, you primarily perform:
- Expansion: This includes multiplying terms within a polynomial, such as opening up \[ (2x-1)^2 = 4x^2 - 4x + 1 \]. Notice how each term within the parenthesis is squared.
- Combining Terms: After expansion, gather like terms with the same variables and exponents together. Simplification results from consolidating terms like \[ 8x^2 + 12x - 8 - 4x^2 + 4x - 1 = 4x^2 + 16x - 9 \].
Common Denominators
A common denominator is crucial for adding or subtracting fractions. It ensures that the terms are compatible for arithmetic operations.
In the exercise:
In the exercise:
- Find the Least Common Denominator (LCD): The LCD is the smallest expression that is evenly divisible by all the denominators in the equation. Here, reaching a common denominator of \[ (x+2)^2 \] allows you to merge terms seamlessly.
- Adjusting Terms: Adjust numerators to reflect the common denominator, as by multiplying the necessary terms accordingly, allowing for \[ \frac{4(2x-1)}{x+2} = \frac{4(2x-1)(x+2)}{(x+2)^2} \].
