Problem 51 - Algebra and Trigonometry

Question

Perform the indicated operations and simplify. $$ (1-b)^{2}(1+b)^{2} $$

Step-by-Step Solution

Answer

\(b^4 - 2b^2 + 1\)

Step 1: Expand (1-b)^2

The expression \((1-b)^2\) is a binomial squared. This can be expanded using the formula \((a-b)^2 = a^2 - 2ab + b^2\). Applying it here:\[(1-b)^2 = 1^2 - 2 \cdot 1 \cdot b + b^2 = 1 - 2b + b^2\].

Step 2: Expand (1+b)^2

Similarly, the expression \((1+b)^2\) is expanded using the same binomial square formula: \((a+b)^2 = a^2 + 2ab + b^2\).Thus, \[(1+b)^2 = 1^2 + 2 \cdot 1 \cdot b + b^2 = 1 + 2b + b^2\].

Step 3: Multiply Expanded Expressions

Now, we multiply the expanded forms of \((1-b)^2\) and \((1+b)^2\).i.e., \((1 - 2b + b^2)(1 + 2b + b^2)\). Use the distributive property (also known as the FOIL method for binomials) to carry out this multiplication: - First: \(1 \times (1 + 2b + b^2) = 1 + 2b + b^2\) - Outer: \(-2b \times (1 + 2b + b^2) = -2b - 4b^2 - 2b^3\) - Inner: \(b^2 \times (1 + 2b + b^2) = b^2 + 2b^3 + b^4\)Combine these results: \[1 + 2b + b^2 - 2b - 4b^2 - 2b^3 + b^2 + 2b^3 + b^4\].

Step 4: Simplify the Expression

Now, combine like terms from the expression:\[1 + (2b - 2b) + (b^2 - 4b^2 + b^2) + (-2b^3 + 2b^3) + b^4\]Simplify to: - \(1 + 0 + (-2b^2) + 0 + b^4\) - Resulting in: \[b^4 - 2b^2 + 1\].

Key Concepts

Binomial Expansion

Binomial expansion is a key concept in algebra that allows us to simplify expressions involving the square or higher powers of binomials. A binomial is an algebraic expression that contains two distinct terms, such as

• \((a - b)\)
• \((x + y)\)

When we square a binomial expression, we employ specific formulas to expand it fully. Here’s a simple breakdown:

• For binomials of the type \((a - b)^2\), use the formula \((a-b)^2 = a^2 - 2ab + b^2\)
• For binomials of the type \((a + b)^2\), use the formula \((a+b)^2 = a^2 + 2ab + b^2\)

Understanding this concept is crucial because it simplifies the process of opening up expressions, such as the ones we encounter in the problem \((1-b)^2\) and \((1+b)^2\). This is akin to unpacking the expression to reveal its full form. It's like breaking down a compound word into two simpler words, making it easier to manipulate in further algebraic operations.

Distributive Property

The distributive property, often referred to informally as the FOIL method when dealing with binomials, is indispensable when multiplying expressions that have been expanded using binomial expansion. This property states that the product of a sum and another term can be distributed to, or multiplied by, each addend in the sum:

• If you have expressions like \((a)(b + c)\), you distribute \(a\) to both \(b\) and \(c\), resulting in a formula that looks like: \(a \cdot b + a \cdot c\).

In algebra, applying the distributive property to expanded expressions such as \((1 - 2b + b^2)\) and \((1 + 2b + b^2)\) requires breaking down each term and multiplying it, step-by-step:

• First, deal with the first terms from each binomial.
• Tackle the outer terms next.
• Progress to the inner terms.
• Finally, focus on the last terms.

These steps result in a comprehensive expansion that showcases each possible multiplication, akin to unfolding a multi-layered origami figure into a flat piece of paper. Mastery of this technique is critical in algebra to ensure precise development and simplification of complex expressions.

Simplifying Algebraic Expressions

Simplifying algebraic expressions involves combining like terms to make an expression clearer and more manageable. Once you've used the binomial expansion and the distributive property to express a polynomial as fully as possible, focus on simplifying. Here’s how you do it effectively:

• Identify like terms: These are terms that contain the same variables raised to the same powers.
• Combine these like terms by adding or subtracting their coefficients while keeping the variables unchanged. For instance, \(2b - 2b = 0\) simplifies by canceling out.

In our example problem: \[1 + 2b + b^2 - 2b - 4b^2 - 2b^3 + b^2 + 2b^3 + b^4\],combining the like terms results in:

• Adding and subtracting like terms to simplify: \(b^4 - 2b^2 + 1\).

This clean results helps clarify the original expression's form, making it much easier to interpret and work with in further mathematical calculations. Understanding how to simplify using these principles is like organizing a cluttered desk - it makes everything easier to find and use.