Problem 34 - Algebra 1: Concepts and Skills

Question

Write the square of the binomial as a trinomial. $$ (x-7)^{2} $$

Step-by-Step Solution

Answer

The square of the binomial (x-7)^{2} written as a trinomial is \(x^{2} - 14x + 49\).

Step 1: Identify a and b

In this given exercise, x is a and -7 is b. This can be understood from the standard form for the binomial which is (a+b)^{2}.

Step 2: Apply the formula

Apply the formula to the binomial (a+b)^2 = a^2 + 2ab + b^2. Substitute a with x and b with -7. That will give us: (x)^{2} + 2*(x)*(-7) + (-7)^{2}.

Step 3: Simplify the expression

Simplify the expression obtained in Step 2. The square of x is \(x^{2}\), 2*x*-7 becomes -14x and the square of -7 becomes 49. So, the simplified expression will be \(x^{2} - 14x + 49\).

Key Concepts

Understanding Binomials

A binomial is an algebraic expression containing exactly two terms separated by a plus or minus sign. The term 'bi' signifies two. Therefore, every binomial follows the structure

• Expression with two terms: such as (a + b) or (a - b)
• Can be a part of addition, subtraction, or multiplication

For example, in the binomial (x - 7), the two terms are x and -7. The operation between them is subtraction. Understanding the structure of a binomial is essential, especially when solving problems involving their squares or multiplying them by other expressions.When squaring a binomial, we expand it into a trinomial. This process involves applying the formula:

• \((a + b)^2 = a^2 + 2ab + b^2\)
• Substitute the values of a and b from your binomial.

Observe how it transforms when the specific values are plugged into the formula and subsequently simplified.

Introduction to Trinomials

A trinomial is an algebraic expression made up of three terms. It comes from the term 'tri,' meaning three, indicating the three distinct parts. Trinomials typically look like this:

• Expression with three terms: such as \(a^2 + 2ab + b^2\)
• These expressions can be formed by squaring binomials.

In our example, the square of the binomial (x - 7)^2 results in the trinomial:\(x^2 - 14x + 49\).This trinomial is derived from applying the perfect square formula, demonstrating how trinomial expressions often originate from simpler structures through expansion and simplification processes.Recognizing patterns in trinomials help with operations such as factoring or further algebraic manipulation, serving as a foundational skill in algebra.

Polynomials at a Glance

Polynomials are broad and versatile algebraic expressions constructed by summing several monomials. These monomials are individual terms consisting of variables raised to powers with coefficients. Key features include:

• Contain two or more terms
• Include different degrees based on the highest power of the variable

While binomials and trinomials are specific types of polynomials, a polynomial of any number of terms follows a similar principle. An example of a polynomial might be:\(3x^3 + 5x^2 - 2x + 7\)This consists of four terms, making it a polynomial of degree three (highest power of the variable is three).Understanding polynomials requires recognizing the parts such as coefficients, exponents, and the constant term. Each of these components plays a critical role in the behavior and properties of the expression. Once you are familiar with trinomials and binomials, operating on more complex polynomials becomes more intuitive.