Problem 29 - Algebra A Combined Function

Question

Multiply. $$ (2 x-1)^{2} $$

Step-by-Step Solution

Answer

The expanded expression of \((2x - 1)^2\) is \(4x^2 - 4x + 1\).

Step 1: Recognize the Pattern

This problem requires expanding a binomial expression squared. Recognize that the expression \((2x - 1)^2\) is of the form \((a - b)^2\) where \(a = 2x\) and \(b = 1\). The formula for the square of a binomial \((a - b)^2\) is \(a^2 - 2ab + b^2\).

Step 2: Apply the Formula

Substitute \(a = 2x\) and \(b = 1\) into the formula \(a^2 - 2ab + b^2\), resulting in \((2x)^2 - 2(2x)(1) + 1^2\).

Step 3: Calculate Each Term

- First term: \((2x)^2 = 4x^2\).- Second term: \(-2(2x)(1) = -4x\).- Third term: \(1^2 = 1\).

Step 4: Combine the Terms

Combine the calculated terms to form the expanded expression: \(4x^2 - 4x + 1\).

Key Concepts

Algebra

Algebra is a fundamental branch of mathematics that deals with symbols and the rules for manipulating these symbols. It enables us to express complex relationships and concepts in a simple and organized manner. In the given exercise, we are dealing with algebraic expressions, namely binomials, and how to expand them. A binomial is composed of two terms, which in this exercise are

• \(2x\)
• \(-1\)

The focus is on understanding how these terms interact when the binomial is squared. This involves recognizing algebraic patterns such as

• \((a - b)^2 = a^2 - 2ab + b^2\)

This formula allows us to expand the binomial expression by following algebraic principles and applying arithmetic operations. By understanding and practicing these rules, you simplify and solve more complex problems later on. Recognizing the structure of algebraic expressions is therefore crucial, as it forms the basis for further calculations in algebra.

Binomial Theorem

The Binomial Theorem provides a systematic method for expanding binomials raised to any power. It gives you a formula that can be applied to simplify what might otherwise be a tedious multiplication task. In our exercise, we specifically used a special case of this theorem: squaring a binomial.
The generalized theorem states that\[ (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k}b^k \]where

• \(n\) is the power
• \(\binom{n}{k}\) refers to the binomial coefficients

However, for

• \((a - b)^2\)

this reduces to the simpler formula we used:

• \(a^2 - 2ab + b^2\)

By understanding the Binomial Theorem, you can handle more complex expressions, including those with larger exponents. This theorem is a powerful tool that helps transform challenging tasks into manageable steps. It's particularly useful not only in pure mathematics but also in fields like probability and statistics.

Polynomial Multiplication

Polynomial multiplication becomes straightforward once you grasp the rules and techniques for multiplying terms. This is essentially what you're doing when expanding a binomial, as seen in our exercise. The goal is to apply multiplication rules to each term of the binomial, systematically obtaining a new polynomial form.
Consider polynomial multiplication as combining every pair of terms. In the case of squaring

• \( (2x - 1)^2 \)

we compute

• \( (2x)(2x) = 4x^2 \)
• \( (2x)(-1) + (-1)(2x) = -4x \)
• \( (-1)(-1) = 1 \)

Once you calculate each part, you combine them to form the polynomial:

• \( 4x^2 - 4x + 1 \)

This process of polynomial multiplication not only applies to binomials but to polynomials with any number of terms. By mastering this, you'll find multiplication becomes a routine and efficient task in working with polynomials.