Question

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

Short answer

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

Step-by-step solution

Identify the Terms

In our case, the binomial is \( (4t-1) \) and it is squared. Here \( a = 4t \) and \( b = 1 \).

Apply the Binomial Square Formula

Apply the formula for the square of a binomial to find the product. Substituting \( a = 4t \) and \( b = 1 \) into the formula \( (a-b)^2 = a^2 - 2ab + b^2 \), we get \( (4t - 1)^2 = (4t)^2 - 2*(4t)*1 + 1^2 \).

Calculate and Simplify

Simplify to get our final answer. \( (4t - 1)^2 = 16t^2 - 8t + 1 \).

Key concepts

Binomial Expression

A binomial expression is a polynomial with two terms that are distinct, usually written in the form of

Algebraic Identity

An algebraic identity is a standard, proven equation that connects different algebraic expressions in a meaningful way and holds true for all possible variable values that are plugged into these expressions. One of the most universally used algebraic identities is the square of a binomial formula. This identity is written as:

• For the sum of two terms: (a + b)^2 = a^2 + 2ab + b^2
• For the difference of two terms: (a - b)^2 = a^2 - 2ab + b^2

These formulae provide shortcuts for algebraic simplification and multiplication without expanding binomials the long way, which saves time and minimizes calculation errors. More complex algebraic identities build upon the simplicity of the binomial square formula, incorporating it as a part in larger structures of polynomial algebra.

Simplifying Algebraic Expressions

Simplifying algebraic expressions means to rewrite them in a more concise and streamlined form while maintaining their original value. This process often involves combining like terms, applying algebraic identities, factoring, and expanding expressions. Simplification makes it easier to understand, solve, or further manipulate algebraic expressions.

In the given exercise, the simplification starts with recognizing a binomial expression being squared. Then, one of the critical algebraic identities, the binomial square formula, is applied to avoid expanding the binomial in a more time-consuming fashion. This process shifts the focus from merely carrying out the multiplication to understanding the underlying property of squaring binomials, which is a significant objective in algebra.