Problem 29 - Algebra 1: Concepts and Skills

Question

Write the square of the binomial as a trinomial. $$ (x+5)^{2} $$

Step-by-Step Solution

Answer

\(x^{2} + 10x + 25\) is the trinomial representation of \((x+5)^{2}\).

Step 1: Identifying the Binomial Components

In this binomial (x+5), 'a' is represented by 'x' and 'b' is represented by '5'.

Step 2: Applying the Square of Binomial Formula

Apply the formula \((a+b)^{2} = a^{2} + 2ab + b^{2}\) by substituting the identified components 'a' and 'b' into the formula: \((x+5)^{2} = x^{2} + 2*(5*x) + 5^{2}\).

Step 3: Solve to Get the Trinomial

Calculate the values to get the trinomial: \(x^{2} + 10x + 25\) is the trinomial.

Key Concepts

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and mathematical operations such as addition, subtraction, multiplication, and division. They are used to represent mathematical situations and relationships in a concise form.
A simple example of an algebraic expression is \(x + 5\), which is the binomial found in our given problem.

• A binomial is an algebraic expression that contains exactly two terms, like \(x + 5\).
• A trinomial is an algebraic expression that contains exactly three terms, such as \(x^2 + 10x + 25\) in our solution.

By understanding these definitions, you can better manipulate algebraic expressions to simplify, solve, or rewrite them. This basis helps when working with polynomial expressions, of which binomials and trinomials are examples.

Polynomials

Polynomials are a specific type of algebraic expression where terms are made up of variable exponents and coefficients. A proper polynomial can have one or more terms, with operations involving addition, subtraction, or multiplication.
The general form of a polynomial is \(a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0\), where \(a_n, a_{n-1}, ..., a_1, a_0\) are coefficients.

• In our exercise, the binomial \((x + 5)\) is squared to create a trinomial \(x^2 + 10x + 25\).
• The polynomial formed here has a degree of 2 because the highest power of the variable \(x\) is squared.

Polynomials can be categorized by their degree and number of terms. A one-term polynomial is termed a monomial (e.g., \(3x^2\)), while a two-term polynomial is a binomial. A three-term expression, like our trinomial, includes three parts when expanded.

Factoring

Factoring is the process of breaking down an expression into its simplest factors, or expressions that can be multiplied to get the original expression. It's a crucial skill in algebra that facilitates solving equations and simplifying expressions.
In the context of our exercise, the trinomial \(x^2 + 10x + 25\) is the result of expanding the squared binomial \((x+5)^2\). If we want to go back, or "factor" this trinomial, we look for expressions that multiply to form it.

• The trinomial \(x^2 + 10x + 25\) factors simply back to \((x+5)\) often referred to as a perfect square trinomial.
• To check this, multiply \(x+5\) by itself to confirm that it expands again to \(x^2 + 10x + 25\).

Understanding factoring helps solve equations and simplify algebraic expressions, as it allows us to view and solve the expressions in their most elementary form.