Problem 66 - Algebra 1: Concepts and Skills

Question

Find the product. $$ (x+5)^{2} $$

Step-by-Step Solution

Answer

\n The product of the binomial \( (x+5)^{2} \) is \( x^{2}+10x+25 \).\n

Step 1: Identify the binomial

The binomial given in the question is \( (x+5) \). This will be considered as \( (a+b) \) in the formula.

Step 2: Identify a, b and calculate according to formula

Here, \( a=x \) and \( b=5 \). Now, replace a and b in the formula \( a^{2}+2ab+b^{2} \) to get the desired result. This gives us \( x^{2}+2*x*5+5^{2} \).

Step 3: Simplify the expression further

Solving the expression further we obtain \( x^{2}+10x+25 \).

Key Concepts

Expanding Binomials

When we talk about expanding binomials, we refer to the process of multiplying out the terms of a binomial, which is an algebraic expression containing two terms. This is often done with the help of the binomial theorem or by using the distributive property. For example, if we have a binomial squared, such as \( (x+5)^2 \) our goal is to express it as a trinomial — a three-term algebraic expression.

In this particular case, the students need to remember the formula \( (a+b)^2 = a^2 + 2ab + b^2 \) to correctly expand the binomial. It's essential to highlight that the middle term, \( 2ab \) reflects the double product of the binomial's individual terms, which is a common stumbling block for beginners. By systematically applying the formula, one ensures accuracy and attends to the multiplicative relationships between binomial terms.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables (like \( x \)), and operations such as addition, subtraction, multiplication, and division. They are the fundamental building blocks of algebra and serve to represent real-world quantities and their relationships. In the context of the provided exercise, \( (x+5)^{2} \) is an algebraic expression involving the variable \( x \), a constant (5), and an exponent (2).

Understanding how to manipulate these expressions is crucial for solving algebra problems. It is not just about applying the formulas but also comprehending the significance and interaction of each term within the expression. When students master the political structure of algebraic expressions, they can solve a wide range of mathematical problems and recognize patterns that are instrumental for more advanced aspects of algebra.

Simplifying Algebraic Expressions

Simplifying algebraic expressions is the process of reducing them to their simplest form. This typically involves combining like terms, which are terms that have the same variables raised to the same power, by adding or subtracting them, as well as multiplying or dividing coefficients. In the step-by-step solution for the exercise concerning the square of the binomial \( (x+5) \), simplification occurs after expanding the expression. The students must add and multiply constants, combine like terms, if any, and present the final answer as \( x^2 + 10x + 25 \).

In this process, there are no like terms to combine, but it's still considered simplification since we transition from an exponentiated binomial to a clearly expressed trinomial. Prompt simplification avoids unnecessary complications in further operations and clarifies the structure of the expression for subsequent problem-solving, which is a key skill for success in algebra.