Problem 71 - Algebra 1: Concepts and Skills

Question

Find the product. $$ (2 a-9 b)^{2} $$

Step-by-Step Solution

Answer

The result of squaring the given binomial is \(4a^2 - 36ab + 81b^2\).

Step 1: Understanding the task

The problem is to calculate the square of the binomial: \( (2a-9b)^2 \). The concept to apply here is the fact that \( (a+b)^2 = a^2 + 2ab + b^2 \)

Step 2: Apply the formula

Apply the formula to the given binomial. It means, replace \(a\) by \(2a\) and \(b\) by \(-9b\). Then, \( (2a - 9b)^2 = (2a)^2 + 2(2a)(-9b) + (-9b)^2 \)

Step 3: Simplify the expression

Perform the operations within the brackets. Thus, the expression becomes \(4a^2 - 36ab + 81b^2\)

Key Concepts

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and at least one arithmetic operation (such as addition, subtraction, multiplication, division, or exponentiation). For instance, in the expression \(2a - 9b\), \(2a\) and \(9b\) are two algebraic terms subtracted from one another. Each term is a product of a constant and a variable:

• \(2a\) consists of the constant 2 multiplied by the variable \(a\).
• \(9b\) consists of the constant 9 multiplied by the variable \(b\).

In the exercise given, we focus on squaring the binomial \(2a - 9b\). Squaring an expression means raising it to the power of two or multiplying the expression by itself. Understanding how algebraic expressions work is essential to perform polynomial operations such as addition, subtraction, and squaring, with ease.

Squaring Binomials

Squaring binomials is a specific type of polynomial operation where we take a binomial (an algebraic expression with two terms) and multiply it by itself. A common formula that aids in squaring binomials is:\[ (a+b)^2 = a^2 + 2ab + b^2 \]
This formula tells us that when we square a binomial, we end up with a trinomial (an expression with three terms) consisting of the square of the first term, twice the product of the two terms, and the square of the second term.
When applying this formula to a binomial with a subtraction, such as \(2a - 9b\), we adjust the formula accordingly:

• The first term \(a\) becomes \(2a\).
• The second term \(b\) becomes \(-9b\).
• Therefore, \((2a - 9b)^2\) is calculated as \( (2a)^2 + 2(2a)(-9b) + (-9b)^2 \).

It's important to handle the negative sign with care as it affects the sign of the middle term in the squared result.

Polynomial Operations

Polynomial operations include various mathematical processes such as adding, subtracting, multiplying, or dividing polynomials. In the context of the given problem, the polynomial operation of interest is multiplication — specifically, the squaring of a binomial.
Squaring a binomial yields a trinomial, which is a polynomial with three terms. After applying the formula for squaring a binomial:\[ (a - b)^2 = a^2 - 2ab + b^2 \]
we then simplify the expression. Simplification involves performing any arithmetic indicated, such as squaring the individual terms and multiplying them accordingly. As seen in the exercise, the simplified expression should be:

• Square of the first term: \( (2a)^2 = 4a^2 \).
• Twice the product of the two terms: \( 2(2a)(-9b) = -36ab \).
• Square of the second term: \( (-9b)^2 = 81b^2 \).

The final result is a polynomial: \(4a^2 - 36ab + 81b^2\). This step-by-step approach to polynomial operations helps ensure accurate calculations and a better understanding of algebra.