Problem 60 - Algebra A Combined Function

Question

Mixed Practice Multiply. $$ (b-2)^{2} $$

Step-by-Step Solution

Answer

The expression \((b-2)^2\) simplifies to \(b^2 - 4b + 4\).

Step 1: Understand the Problem

The problem requires us to square the binomial expression \((b - 2)\). This involves multiplying the binomial \((b - 2)\) by itself.

Step 2: Set Up the Multiplication

Write the expression as a multiplication problem: \((b - 2) \times (b - 2)\). This indicates we are performing a binomial expansion.

Step 3: Apply the Distributive Property

Use the distributive property to expand the expression: - First, multiply \(b\) by both terms in the second parenthesis: \(b \times b + b \times (-2)\).- Then, multiply \(-2\) by both terms in the second parenthesis: \(-2 \times b + (-2) \times (-2)\).

Step 4: Perform the Multiplications

Calculate each multiplication:- \(b \times b = b^2\)- \(b \times (-2) = -2b\)- \(-2 \times b = -2b\)- \(-2 \times (-2) = 4\)

Step 5: Combine Like Terms

Add the results from step 4:\(b^2 - 2b - 2b + 4\).Combine the like terms \(-2b - 2b\), which results in \(-4b\).

Step 6: Write the Final Expression

The expanded and simplified expression of \((b - 2)^2\) is:\(b^2 - 4b + 4\).

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that allows us to multiply a single term by each term inside a set of parentheses. It's often used when dealing with expressions involving parentheses, ensuring every part of the equation is expanded correctly.

To illustrate, take the binomial \( (b - 2)^2 \). To expand this, we rewrite it as \( (b-2) imes (b-2) \), and apply the distributive property in steps:

• Multiply the first term in the first binomial with each term in the second binomial.
• Multiply the second term in the first binomial with each term in the second binomial.

So, \( b \) is multiplied by both \( b \) and \( -2 \), producing \( b^2 \) and \( -2b \). Then, \( -2 \) is multiplied in the same way, yielding \( -2b \) and \( 4 \).

This approach ensures that no terms are forgotten, making it a reliable method for expanding binomials and polynomials.

Algebraic Multiplication

Algebraic multiplication involves multiplying algebraic expressions to simplify or expand them. This is key in binomial expansions such as \( (b - 2)^2 \).

In our example, multiplying \( (b-2) imes (b-2) \) means performing set muliplications:

• Multiply \( b \) by \( b \) to get \( b^2 \).
• Multiply \( b \) by \( -2 \) to get \( -2b \).
• Multiply \( -2 \) by \( b \) to again get \( -2b \).
• Finally, multiply \( -2 \) by \( -2 \) to get \( 4 \).

This systematic approach uses basic multiplication rules to ensure accurate expansion. Each part of the binomials is multiplied together, which is essential in building up to more complex algebra problems. Being able to handle these multiplications ensures robust algebraic skills.

Combining Like Terms

Combining like terms is a crucial process in simplifying algebraic expressions. It involves adding or subtracting terms that have the same variable raised to the same power. After expanding the expression \( (b - 2)^2 \), we obtain several terms, which need to be combined to simplify the expression.

From our multiplication results, we get:

• \( b^2 \)
• \( -2b \)
• \( -2b \)
• \( 4 \)

Look for terms with the same variables and exponents. Here, \( -2b \) and \( -2b \) are like terms, so they are combined to form \( -4b \).

Ultimately, the simplified expression is \( b^2 - 4b + 4 \). This process allows for clearer and more concise expression results, making it easier to understand and further manipulate these results in algebraic problems.