Problem 23 - Algebra and Trigonometry

Question

Perform the indicated operations and simplify. $$ x^{2}\left(2 x^{2}-x+1\right) $$

Step-by-Step Solution

Answer

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

Step 1: Distribute the Exponent

Distribute the term \(x^2\) across each term in the parentheses. This means you multiply \(x^2\) by each term inside the parentheses: \(2x^2\), \(-x\), and \(1\).

Step 2: Multiply Each Term

1. Start by multiplying \(x^2\) by \(2x^2\): \[ x^2 \cdot 2x^2 = 2x^4 \] 2. Next, multiply \(x^2\) by \(-x\): \[ x^2 \cdot (-x) = -x^3 \] 3. Finally, multiply \(x^2\) by \(1\): \[ x^2 \cdot 1 = x^2 \]

Step 3: Combine All Terms

Combine all the resulting expressions from Step 2:\[ 2x^4 - x^3 + x^2 \]

Step 4: Simplify the Expression

Arrange terms in descending order of the powers (this is already done). Thus, the expression \(2x^4 - x^3 + x^2\) is the simplified form.

Key Concepts

Distributive Property

The Distributive Property is a fundamental concept in algebra, especially when dealing with polynomial multiplication. In simple terms, this property allows you to multiply a single term by each term inside a set of parentheses. What's happening here is like distributing a set of items to several groups one by one. This property can be seen in the exercise where we have to distribute the term \(x^2\) with each term inside the parentheses, \(2x^2 - x + 1\).

• First, distribute \(x^2\) to \(2x^2\).
• Second, multiply \(x^2\) with \(-x\).
• Finally, distribute \(x^2\) to \(1\).

By distributing each term individually, you maintain the equality of the expression throughout the multiplication process. It ensures that no term is left unmultiplied, resulting in an accurate and complete polynomial expression that can then be simplified by combining like terms, if any exist.

Exponent Rules

Exponent rules help us handle powers of variables effectively. Exponents dictate how many times a base is multiplied by itself, and the rules help simplify expressions with exponents.In our exercise, these rules mainly revolve around multiplying exponents. Whenever we multiply terms with the same base, we add their exponents together. Here’s a breakdown:

• For example, in \(x^2 \times 2x^2\), since both terms have the base \(x\), we add their exponents: \(x^{2+2} = x^4\).
• When multiplying \(x^2\) by \(-x\), you adjust the power: \(x^2 \times x^1 = x^{2+1} = x^3\). Remember, \(-x\) means the coefficient is negative, but the base remains the same, which impacts the sign, not the exponents.

By understanding and applying exponent rules, you simplify the polynomials systematically. This approach is crucial for maintaining accuracy and ensuring that the coefficients and powers align correctly as the expressions evolve.

Polynomial Simplification

Polynomial simplification involves arranging and combining terms to present the expression in its simplest form. This process ensures that your solution is both elegant and correct.After applying the distributive property and utilizing exponent rules, you gather the results into one expression. Each term should be ordered by the degree, typically starting with the highest power. In our specific example, the polynomial is already organized from high to low: \(2x^4 - x^3 + x^2\).

• Order the terms based on their exponents, descending from left to right.
• Combine like terms, though in our case, they are already simplified since no similar degree terms exist.

The goal of simplification is to create an expression that's easy to read and clearly represents the solution. This method allows for complex expressions to be conveyed in a manageable and straightforward manner, which is especially helpful when further calculations or evaluations are to follow.