Problem 67 - Algebra 1: Concepts and Skills

Question

Find the product. $$ (11-6 x)^{2} $$

Step-by-Step Solution

Answer

The product of \( (11-6x)^{2} \) is \( 121 - 132x + 36x^2 \).

Step 1: Identifying Terms

In the expression \( (11-6x)^{2} \), identify the 'a' term and the 'b' term. Here, 'a' is 11 and 'b' is \( 6x \).

Step 2: Expanding the Square

Apply the formula \( (a-b)^{2} = a^2 - 2ab + b^2 \) substituting 'a' and 'b' with 11 and \( 6x \) respectively. This results in \( (11)^2 - 2*11*6x + (6x)^2 \).

Step 3: Solving the Equation

Now simplify the equation. \( (11)^2 = 121 \). The middle term equals \( 2 * 11 * 6x = 132x \). And \( (6x)^2 = 36x^2 \). So, the expanded form of the original expression becomes \( 121 - 132x + 36x^2 \).

Key Concepts

Square of a Binomial

When we talk about the square of a binomial, we're exploring expressions like \((a - b)^2\) or \((a + b)^2\), where you have two terms inside a parenthesis raised to the power of 2. In this particular example, we're focusing on \((11 - 6x)^2\). The square of a binomial follows a specific formula that makes it easier to expand:

• \((a + b)^2 = a^2 + 2ab + b^2\)
• \((a - b)^2 = a^2 - 2ab + b^2\)

In our case, the expression is \((11 - 6x)^2\), so it matches the second formula, where 'a' is 11, and 'b' is \(6x\). This means:- \(a^2\) translates to \((11)^2\),- \(-2ab\) becomes \(-2 \times 11 \times 6x\),- \(b^2\) is \((6x)^2\).
Each part of this formula helps us simplify and expand the expression step by step.

Algebraic Simplification

Algebraic simplification is the process of breaking down complex expressions into simpler ones. Once we have applied the binomial square formula, we need to simplify the expression further. For \((11 - 6x)^2\), after substituting the values in, you get:- \((11)^2 = 121\)- \(-2 \times 11 \times 6x = -132x\)- \((6x)^2 = 36x^2\)
This gives you an expression: 121 - 132x + 36x^2.
Simplifying isn't only about calculating, but ensuring each term is correctly derived and combined to form a polynomial. It's essential because it reduces the possibility of errors and helps in better understanding.

Polynomial Expansion

Polynomial expansion refers to the process of turning a compact expression like \((11 - 6x)^2\) into a longer, more detailed expression, which is a polynomial, through multiplication and combination of terms. This particular process involves multiplying each part of the binomial by itself according to its formula.
A polynomial consists of several terms, each a product of numbers and variables raised to powers. After expanding \((11 - 6x)^2\) using the formula \(a^2 - 2ab + b^2\), you obtain the expanded polynomial: 121 - 132x + 36x^2.
This polynomial is in standard form: starting with the constant term, followed by the 'x' term, and ending with the 'x^2' term, each part derived directly from the step-by-step building block expansion of the binomial.