Problem 34 - Algebra and Trigonometry

Question

Find the sum, difference, or product. \(v^{3}(v-9)-2 v^{2}(2-2 v)\)

Step-by-Step Solution

Answer

The simplified expression is \(v^4 - 5v^3 - 4v^2\).

Step 1: Distribute Terms in Each Expression

First, let's distribute the terms in each part of the expression.- For the first term, distribute: \(v^3(v - 9) = v^4 - 9v^3\) - For the second term, distribute: \(-2v^2(2 - 2v) = -2v^2 \cdot 2 + 2v^2 \cdot 2v = -4v^2 + 4v^3\)

Step 2: Combine Like Terms

Now, combine the like terms from the expanded expressions:We have: \(v^4 - 9v^3 - 4v^2 + 4v^3\).Combine the terms for \(v^3\):\(-9v^3 + 4v^3 = -5v^3\).Thus, the expression simplifies to:\(v^4 - 5v^3 - 4v^2\).

Step 3: Final Simplified Expression

The final simplified expression, after combining all the like terms, is:\(v^4 - 5v^3 - 4v^2\).

Key Concepts

Distributive Property

The distributive property is a fundamental concept that allows you to simplify expressions by distributing a term across terms inside parentheses. This is particularly useful when dealing with polynomial expressions, where terms are often multiplied by variables or other terms. To apply the distributive property, multiply each term inside the parentheses by the term outside. For example, in the expression \(v^3(v - 9)\), you multiply \(v^3\) by both \(v\) and \(-9\), giving \(v^4 - 9v^3\). Similarly, for \(-2v^2(2 - 2v)\), multiply \(-2v^2\) by both \(2\) and \(-2v\), resulting in \(-4v^2 + 4v^3\).Always remember these key steps:

• Identify the term to be distributed across the parentheses.
• Multiply the outside term with each term inside.
• Keep track of positive and negative signs.

Understanding this property is crucial for simplifying complex expressions efficiently.

Like Terms

When working with polynomials, it's essential to recognize and combine like terms. Like terms have the same variable raised to the same power. This means terms like \(v^3\) and \(-9v^3\) can be combined, while terms like \(v^2\) and \(v^4\) cannot.In the example from the solution, after distributing the terms, the expression \(v^4 - 9v^3 - 4v^2 + 4v^3\) contains like terms \(-9v^3\) and \(4v^3\). These can be combined by adding their coefficients, leading to \(-5v^3\). To effectively combine like terms, follow these tips:

• Identify terms with identical variables and exponents.
• Add or subtract their coefficients accordingly.
• Write the simplified term while maintaining the variable and its power.

This process not only simplifies the expression but also makes it more manageable for further operations.

Polynomial Simplification

Polynomial simplification is the process of making a polynomial expression as concise and readable as possible. This involves distributing terms, combining like terms, and arranging them in order of decreasing powers.The simplified expression \(v^4 - 5v^3 - 4v^2\) is achieved by applying both the distributive property and combining like terms. Notice that the terms are arranged in descending order of power: \(v^4\), \(-5v^3\), and \(-4v^2\). Key aspects to remember for simplification:

• Use the distributive property to remove parentheses and expand the expression.
• Identify and combine like terms for a more concise form.
• Arrange terms from highest to lowest powers to follow standard polynomial form.

Practicing these steps ensures clarity and simplicity, which are crucial in further algebraic manipulations and in understanding polynomials.