Problem 65 - Elementary Algebra

Question

For the following problems, note how many: $$ 2 \text { 's in } 2 x^{2}(x-7) ? $$

Step-by-Step Solution

Answer

Answer: There are 2 occurrences of 2 in the given expression.

Step 1: Expand the Expression

We need to expand the given expression \(2x^2(x-7)\). To do so, we will distribute the \(2x^2\) term within the brackets: \(2x^2(x-7) = 2x^2\cdot x - 2x^2\cdot7\)

Step 2: Simplify the Expression

Now, let's simplify the expression obtained in step 1: \(2x^2\cdot x - 2x^2\cdot7 = 2x^3 - 14x^2\)

Step 3: Count the Number of 2's

In the simplified expression, \(2x^3 - 14x^2\), there are two terms and both have a 2 as their coefficient. Therefore, there are 2 occurrences of 2 in the given expression.

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra, allowing us to multiply a single term by each term within parenthesis. It forms the backbone for expanding expressions like the one in our exercise. In this problem, we distributed the term \(2x^2\) to both \(x\) and \(-7\), effectively allowing us to tackle each part of the expression individually.

Here’s how it works:

• Multiply the term outside the parenthesis by the first term inside: \(2x^2 \times x = 2x^3\).
• Then multiply the same term by the second term inside: \(2x^2 \times -7 = -14x^2\).

This property is especially useful in algebra for simplifying expressions and solving equations. By using the distributive property, we avoid dealing with more complex operations too early in the process.

Remember, the distributive property applies to subtraction as well, as seen when we distributed over \(x-7\). This flexibility is what makes it a powerful tool in algebra.

Coefficients in Algebra

Coefficients are the numerical factors in terms of an algebraic expression, representing the "amount" each variable is multiplied by. In our exercise, after expanding \(2x^2(x-7)\), we reached the expression \(2x^3 - 14x^2\). The numbers 2 and 14, in this context, serve as coefficients.

It's vital to recognize coefficients as they often provide clues on how to scale or balance equations. In this example:

• The coefficient 2, in \(2x^3\), highlights how many times \(x^3\) appears as a result of the multiplication.
• The coefficient -14, in \(-14x^2\), indicates its negative contribution and how many times \(x^2\) subtracts from the expression.

Understanding coefficients helps in identifying patterns and relationships in algebra, ultimately leading to more efficient problem-solving. It’s noteworthy that specific coefficients were primary in answering how many "2's" were present in the expression.

Polynomial Simplification

Polynomial simplification is the process of making an expression as simple as possible by combining like terms and applying arithmetic operations. This prevents the equation from becoming overly complicated and easier to understand. Upon distributing \(2x^2\) in our problem, we obtained \(2x^3 - 14x^2\).

Simplifying a polynomial like this includes several key actions:

• Combine like terms: Ensure all terms with the same degree are added or subtracted accurately.
• Follow arithmetic rules: Handle negative signs and coefficients correctly.

While our problem didn't require further simplification beyond redistribution of terms, typically, authors simplify further to present an expression in its most reduced form.

Simplified polynomials ease further manipulation in algebra, such as solving equations or finding roots. By reducing steps of computation or combining terms effectively, polynomial simplification fosters a deeper understanding of the expression’s structure.