Problem 13 - Algebra Form and Function

Question

In Exercises \(13-18,\) is the expression a polynomial in the given variable? $$ \left(4-2 p^{2}\right) p+3 p-(p+2)^{2}, \text { in } p $$

Step-by-Step Solution

Answer

Answer: Yes, the given expression is a polynomial in variable p. The simplified expression is \(-2p^3 - p^2 + 3p\).

Step 1: Expanded and simplify the expression

First, we expand and simplify the given expression: \((4-2p^2)p+3p-(p+2)^2\)

Step 2: Expand the brackets

Next, we expand the brackets and simplify: \(4p - 2p^3 + 3p - (p^2 + 4p + 4)\)

Step 3: Combine similar terms

Now, we combine the similar terms: \(4p - 2p^3 + 3p - p^2 - 4p - 4\)

Step 4: Simplify

Lastly, we simplify the expression by adding and subtracting the coefficients of the terms with the same power of p: \(-2p^3 - p^2 + 3p\)

Step 5: Checking if the expression is a polynomial

Now that we have the simplified expression, we need to check if it is a polynomial. A polynomial in variable p is a mathematical expression consisting of a sum of terms, each term including a variable 'p' raised to a non-negative integer power. The simplified expression is: \(-2p^3 - p^2 + 3p\) This expression is a sum of terms where each term has p raised to a non-negative integer power. Therefore, the given expression is a polynomial in variable p.

Key Concepts

Algebra

Algebra is one of the fundamental branches of mathematics focused on using letters and symbols to represent numbers and quantities in formulas and equations. It involves various operations and the manipulation of expressions to solve problems. In algebra, expressions can consist of numbers, variables, and mathematical operations such as addition, subtraction, multiplication, and division. Greenfield algebra aims to simplify complex expressions and solve equations effectively.When dealing with polynomials, which are specific types of algebraic expressions, certain rules and techniques come into play. The algebraic structure is determined by how terms are arranged and operated upon. This ensures that each term follows the standard form, making it easier to analyze and solve.Algebraic manipulation is crucial when handling expressions like \[(-2p^3 - p^2 + 3p)\], as it provides the skills required to rearrange, simplify, and ultimately define if they meet certain mathematical criteria, such as determining if an expression qualifies as a polynomial.

Variable Manipulation

Variable manipulation refers to the process by which mathematical expressions involving variables are simplified or altered through algebraic rules. In the expression \((4-2p^2)p+3p-(p+2)^2\), the variable 'p' is manipulated through expansion, collection of like terms, and simplification.

• **Expansion** - This involves multiplying out brackets to express polynomial terms fully, as seen in transforming \((p+2)^2\) into \(p^2 + 4p + 4\).
• **Combining Like Terms** - Similar terms such as \(4p\) and \(3p\) are combined to form a simplified result like \(7p\).
• **Simplification** - Variables are manipulated to form the simplest version of the expression, enabling easier interpretation and solution. For instance, \(-2p^3 - p^2 + 3p\), where terms are arranged in decreasing order of power of 'p'.

By mastering variable manipulation, students can efficiently solve algebraic expressions, detect patterns, and identify polynomial structures within complex equations.

Polynomial Format

A polynomial format is a mathematical expression composed of variables raised to non-negative integer powers, combined with coefficients. These expressions are typically written from highest to lowest power of the variable.In more detail, a polynomial in one variable—like polynomial in 'p'—will appear as a sum of terms such as:\[-2p^3 - p^2 + 3p\]Here’s how polynomial formats are characterized:

• Each term comprises a variable (like 'p') raised to a power (like 3 in \(-2p^3\)).
• Coefficients are the numerical parts of each term, such as -2 in \(-2p^3\).
• The powers are non-negative integers, which maintains the definition of a polynomial.

Checking if an expression is in polynomial format involves ensuring all exponents are whole numbers and that there is no division by a variable, ensuring smooth analysis and simpler calculus applications.