Problem 63 - Algebra 1: Concepts and Skills

Question

Find the product. $$ (4 t-1)^{2} $$

Step-by-Step Solution

Answer

The product is \(16t^2 - 8t + 1\).

Step 1: Identify the values of a and b

From the given binomial \((4t - 1)\), a is the term '4t' and b is '1'.

Step 2: Square a and b

Square the values of a and b. Here, \(a^2 = (4t)^2 = 16t^2\) and \(b^2 = 1^2 = 1\).

Step 3: Multiply a and b and double the result

Next, multiply a and b and then double the result to calculate \(2ab\). So, \(2ab = 2*4t*1 = 8t\).

Step 4: Add the results

Finally, substitute the values calculated earlier into the formula of the square of a binomial. Now, \((a - b)^2 = a^2 - 2ab + b^2 = 16t^2 - 8t + 1\).

Key Concepts

Binomial Multiplication

When it comes to binomial multiplication, it's essential to understand that a binomial is a polynomial with two terms. To multiply a binomial by itself, which is squaring the binomial, you apply the formula \(a + b)^2 = a^2 + 2ab + b^2\) or \(a - b)^2 = a^2 - 2ab + b^2\), depending on the sign between the terms. This method is also known as the square of a binomial.

The exercise \( (4t - 1)^2 \) involves squaring the binomial \(4t - 1\). Follow these steps:

• Start by identifying the two terms in the binomial, often represented as \(a\) and \(b\).
• Square each term independently: \(a^2\) and \(b^2\).
• Multiply the original terms (considering their sign) and double the result for the middle term, resulting in \(2ab\).
• Add these values together to conclude the simplification.

By following these steps correctly, students can expand and simplify any binomial squares they encounter in their algebraic exercises.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables (like \(t\)), and arithmetic operations (addition, subtraction, multiplication, and division). In the context of the binomial \(4t - 1\), we have an algebraic expression consisting of a variable term \(4t\), a constant term \(1\), and a subtraction operation.

In algebra, understanding how to manipulate these expressions is crucial for problem-solving. When you square the binomial \(4t - 1\), you are performing an operation on the entire algebraic expression, not just on a single term, which illustrates the importance of dealing with each term correctly. The aim is to preserve the relationships between the terms through proper application of algebraic rules.

Simplifying Algebraic Expressions

Simplifying algebraic expressions is a fundamental skill in algebra. It involves reducing the expression to its simplest form while maintaining its value. When working with binomial squares, simplifying involves expanding and combining like terms in a systematic way.

For the binomial \(4t - 1\^2\), simplification goes through these key points:

• Square each term in the binomial (the \(a^2\) and \(b^2\) in the formula).
• Calculate the \(2ab\) term by multiplying the original terms and doubling them, which results in the middle term of your expanded expression.
• Combine these calculated values based on the appropriate binomial square formula to get the simplest form of your expression.

Throughout this process, careful arithmetic and adherence to algebraic principles help ensure accuracy and a correct final answer.