Problem 35 - Algebra and Trigonometry

Question

Find the sum, difference, or product. \(x^{2}\left(2 x^{2}-x+1\right)\)

Step-by-Step Solution

Answer

The expanded expression is \(2x^4 - x^3 + x^2\).

Step 1: Understand the Problem

We need to expand the expression \(x^2(2x^2 - x + 1)\). This involves multiplying each term inside the parentheses by \(x^2\).

Step 2: Distribute \(x^2\) Through the Parentheses

Multiply \(x^2\) with each term inside the parentheses: 1. \(x^2 \times 2x^2 = 2x^4\)2. \(x^2 \times -x = -x^3\)3. \(x^2 \times 1 = x^2\).

Step 3: Combine the Products

Combine the products obtained from the distribution into a single polynomial expression: \(2x^4 - x^3 + x^2\).

Key Concepts

Distributive Property

Distributive property is a fundamental principle in algebra, helping us simplify expressions involving terms inside and outside parentheses. It states that if you have an expression of the form \( a(b+c) \), you can "distribute" \( a \) to both \( b \) and \( c \), which results in \( ab + ac \). This rule is not limited to numbers but can also be applied to algebraic expressions with variables.
For example, in the exercise where we expand \( x^2(2x^2 - x + 1) \), the distributive property allows us to multiply \( x^2 \) by each term inside the parentheses individually:

• \( x^2 \times 2x^2 = 2x^4 \)
• \( x^2 \times -x = -x^3 \)
• \( x^2 \times 1 = x^2 \)

This approach systematically breaks down the expression and rearranges it into a simplified form. This property is crucial for tackling more complex algebraic expressions.

Multiplication of Polynomials

Multiplying polynomials involves using the distributive property to combine each term in one polynomial with each term in another. When you have a polynomial multiplied by a monomial, like \( x^2 \) and \( 2x^2 - x + 1 \), you multiply the monomial by each term in the polynomial. This not only helps in expanding and simplifying polynomials but also in polynomial factoring and solving polynomial equations.
Polynomials are made up of terms which are variables raised to a power, often added or subtracted from one another, for example, \( 2x^2 \), \( -x \), and \( 1 \) in the original expression. By applying the multiplication step-by-step, it becomes more straightforward to handle different powers of variables systematically.
When dealing with polynomials of more than one term, always ensure each term is correctly multiplied to maintain the operation's accuracy, leading us to form a new polynomial like the expanded expression \( 2x^4 - x^3 + x^2 \).

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations that represent a specific quantity. They provide a way to generalize mathematical relationships and constraints.
An expression like \( x^2(2x^2 - x + 1) \) serves as a compact way to denote more complex equations or properties using symbols. The process shown in the exercise is typical for manipulating these expressions to uncover more insights or solutions, like finding sums, differences, or products.
Understanding the components of algebraic expressions such as coefficients (numerical parts like 2 in \( 2x^2 \)), variables (like \( x \)), and exponents (the power to which a variable is raised, such as the 2 in \( x^2 \)) is crucial. Recognizing how these interact through operations like distribution and multiplication gives you the tools to simplify, evaluate, or reformulate the expressions into more usable forms.