Problem 48 - Elementary Algebra

Question

For the following problems, find the products. $$ (f-1.006)^{2} $$

Step-by-Step Solution

Answer

Question: Expand and simplify the binomial expression \((f-1.006)^{2}\). Answer: \((f-1.006)^{2} = f^{2} - 2.012f + 1.012036\)

Step 1: Identify a and b

First, identify the values of \(a\) and \(b\) in the given binomial expression: \((f-1.006)^{2}\). Here, \(a = f\) and \(b = -1.006\).

Step 2: Apply the binomial expansion formula

Now, apply the binomial expansion formula \((a+b)^{2}=a^{2}+2ab+b^{2}\) with \(a = f\) and \(b = -1.006\). Remember to keep the negative sign for \(b\): $$(f-1.006)^{2} = f^{2} + 2(f)(-1.006) + (-1.006)^{2}.$$

Step 3: Calculate the individual terms

Next, calculate each term in the expanded expression: 1. \(f^2\) 2. \(2(f)(-1.006) = -2.012f\) 3. \((-1.006)^2 = 1.012036\)

Step 4: Combine the terms

Finally, combine the three terms from Step 3 to find the product of the given binomial expression: $$(f-1.006)^{2} = f^{2} - 2.012f + 1.012036$$

Key Concepts

Algebraic Expressions

Algebraic expressions are a fundamental concept in mathematics that involve combining numbers, variables, and operations. These expressions are like sentences in math, where numbers are the words and operations like addition and subtraction act like conjunctions joining them together. They can represent a wide variety of mathematical ideas and relationships.

These expressions often include

• Constants (specific numbers such as 1, 2.5, -0.75)
• Variables (letters that stand for unknown values, like \(x\) or \(y\))
• Operators (mathematical actions such as \(+\), \(-\), \(\times\), \(\div\))

The expression \((f-1.006)^2\) is an algebraic expression where \(f\) is a variable and \(-1.006\) is a constant. The operations involved are subtraction, indicated by the minus sign, and squaring, indicated by the power of 2.

Understanding how to manipulate these algebraic expressions by applying operations is essential. It allows you to simplify complex expressions and solve equations effectively. This skill is crucial for higher-level math and real-world applications, such as physics, engineering, and computer science.

Quadratic Expressions

Quadratic expressions are a type of polynomial that includes terms with a degree of two. In simple terms, when expressions like \((a+b)^2\) are expanded, they form what we call quadratic expressions. A typical form for a quadratic in one variable is \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants with \(a eq 0\).

When working with an expression like \((f-1.006)^2\), we can transform it into its quadratic form by expanding it. By using the formula \((a-b)^2 = a^2 - 2ab + b^2\), we identify:

• \(a = f\)
• \(b = 1.006\)

Applying these to the formula results in:
\(f^2 - 2(f)(1.006) + (1.006)^2\).

This expanded form \(f^2 - 2.012f + 1.012036\) is a typical quadratic expression because it clearly contains a squared term \(f^2\), along with linear \(f\) and constant terms. Quadratics are fundamental to solving many real-world problems, from calculating areas to modeling physical phenomena.

Polynomial Multiplication

Polynomial multiplication is an essential process that involves multiplying together terms of polynomials. This process is vital for expanding expressions like \((f-1.006)^2\). In simple terms, it means distributing each term in one polynomial to every term in the other. It breaks down to the distributive property in arithmetic where \( a(b + c) = ab + ac \).

In practice, multiplying the binomial \((f - 1.006)\) by itself requires you to:

• Multiply the first terms: \(f \times f = f^2\)
• Multiply the outer terms: \(f \times -1.006 = -1.006f\)
• Multiply the inner terms: \(-1.006 \times f = -1.006f\)
• Multiply the last terms: \(-1.006 \times -1.006 = 1.012036\)

Then, combine like terms, specifically by adding \(-1.006f\) and \(-1.006f\) to get \(-2.012f\).

This results in the expanded expression: \(f^2 - 2.012f + 1.012036\). Mastering polynomial multiplication helps in simplifying expressions and solving higher-degree polynomial equations. It's used in areas like electronic circuit designs and economic models, where such intricate calculations are common.