Problem 45 - Elementary Algebra

Question

For the following problems, find the products. $$ (x-3.1)^{2} $$

Step-by-Step Solution

Answer

Answer: The expanded form of the expression \((x-3.1)^2\) is \(x^2 - 6.2x + 9.61\).

Step 1: Identify the formula

In this problem, we will be using the formula for the square of the difference: \((a-b)^2 = a^2 - 2ab + b^2\). This will allow us to expand the expression \((x-3.1)^2\).

Step 2: Substitute the values

Substitute \(a = x\) and \(b = 3.1\) into the formula: \((x-3.1)^2 = (x-3.1)(x-3.1) = x^2 - 2(x)(3.1) + (3.1)^2\)

Step 3: Calculate the products

Calculate each product individually: \(x^2 - 2(x)(3.1) + (3.1)^2 = x^2 - 6.2x + 9.61\)

Step 4: Write the final expression

The final expression for the product is: \((x-3.1)^2 = x^2 - 6.2x + 9.61\)

Key Concepts

Square of a Binomial

Expanding a binomial like \((x - 3.1)^2\) might seem slightly intimidating at first, but it's straightforward once you learn the formula. The square of a binomial refers to multiplying the binomial by itself. For any binomial \((a-b)\), squaring it means calculating \((a-b)^2\).There is a specific formula for this: \((a-b)^2 = a^2 - 2ab + b^2\).This formula allows you to avoid repetitive multiplication and quickly find the expansion.
For instance, when squaring \((x - 3.1)\), set \(a = x\) and \(b = 3.1\).

• First, square \(a\) to get \(x^2\).
• Next, the product of \(2ab\) becomes \(-6.2x\), as \(-2(x)(3.1)\).
• Lastly, square \(b\) to achieve \(9.61\).

By using this formula, you turn any binomial square into a simple polynomial.

Algebraic Formulas

Algebraic formulas are like powerful tools in mathematics that help simplify and solve complex expressions. By knowing these formulas, you can make quick work of what might first glance seem like daunting tasks.The formula for the square of a binomial, such as \((a - b)^2 = a^2 - 2ab + b^2\),is one of these essential algebraic tools. They provide a convenient shortcut to manually expanding and help prevent common calculus or arithmetic errors.
These formulas are not just theoretical; they streamline real problems faced in everyday homework.
Understanding and memorizing such formulas allow students to efficiently handle multiple problems without bombarding themselves with unnecessary calculations.

• Recognize the patterns in different formulas.
• Apply them correctly based on the given expression.
• Practicing repeatedly will cement this tool in your math toolkit.

Always remember, formulas are your friends—reliable, quick, and accurate.

Polynomial Expansion

Polynomial expansion involves turning a condensed expression into a larger one,where each term in the product results in a polynomial term.In this case, expanding \((x-3.1)^2\)results in three terms: \(x^2 - 6.2x + 9.61\).This process begins with identifying the polynomial form and involves using algebraic formulas.

• The purpose is to reveal the underlying structure of the expression.
• Here, the starting binomial is expanded into a sum of terms.
• Each term provides an insight into the expression's behavior, i.e., how changes in \(x\) affect the polynomial.

Expanding polynomials is a vital skill in algebra. It allows for simplifying and solving expressions, provides insight into function behavior, and is foundational for calculus and higher mathematics. Whether in a simple or complex setting, understanding the steps to expand can lead to deeper mathematical insights.