Problem 58 - Algebra 1: Concepts and Skills

Question

Find the product. $$ (7 x-11)^{2} $$

Step-by-Step Solution

Answer

The product is \(49x^2 - 154x + 121\).

Step 1: Identify the first term a and the second term b

In this expression, the first term 'a' is '7x' and the second term 'b' is '11'.

Step 2: Apply the Binomial Theorem

Apply the binomial theorem, recognizing that the binomial has to be squared: \((a-b)^2 = a^2 - 2ab + b^2\)

Step 3: Substitute and Simplify

Substitute '7x' for 'a' and '11' for 'b' in the formula, which gives: \((7x)^2 - 2*(7x)*11 + (11)^2\). Then simplify to: \(49x^2 - 154x + 121\)

Key Concepts

Polynomial

Polynomials are algebraic expressions that involve sums and differences of terms. Each term is a product of a constant and a variable raised to a non-negative integer power. In other words, polynomials are expressions like \(3x^2 + 2x - 5\), where each term has a well-defined degree. The degree of a polynomial is determined by the term with the highest exponent.

Polynomials play a crucial role in mathematics because they can model a variety of real-world situations. They are easy to manipulate, and operations such as addition, subtraction, multiplication, and finding roots are fundamental to algebra. Understanding the structure of polynomials helps in solving many algebraic expressions effectively.

Algebraic Expressions

Algebraic expressions consist of numbers, variables, and operations (such as addition, subtraction, multiplication, and division) combined in a meaningful way. An algebraic expression does not include an equality sign; if it does, it's considered an equation.

There are different types of algebraic expressions, including:

• Monomial: An algebraic expression that's a single term, like \(3x\).
• Binomial: An expression with two terms, such as \(7x - 11\).
• Polynomial: An expression with one or more terms.

Understanding how to manipulate and simplify these expressions is essential to algebra. It allows you to solve problems by making complex expressions easier to work with.

Squaring Binomials

Squaring a binomial means multiplying a binomial by itself. The process utilizes the Binomial Theorem, which simplifies the calculation by giving a direct formula. For any binomial \((a - b)^2\), the expanded form is \(a^2 - 2ab + b^2\). This formula comes from applying the distributive property and simplifies what could otherwise be a cumbersome calculation.

Applying this to the expression \((7x - 11)^2\), we follow these steps:

• Square the first term: \((7x)^2 = 49x^2\).
• Multiply the first term by the second term, multiply by 2: \(-2 * (7x) * 11 = -154x\).
• Square the second term: \((11)^2 = 121\).

Combining these results gives the simplified expression: \(49x^2 - 154x + 121\). Knowing how to square binomials simplifies many algebraic processes and is a key step in solving advanced algebraic expressions.