Problem 19 - Algebra 1: Concepts and Skills

Question

Tell whether the expression is the square of a binomial. $$ y^{2}-10 y+25 $$

Step-by-Step Solution

Answer

Yes, the expression is the square of a binomial \((y - 5)^2\).

Step 1: Identify the components

First, identify the components of the given equation; \(a^2\) corresponds to \(y^2\), \(2ab\) corresponds to \(-10y\), and \(b^2\) corresponds to \(25\). From \(a^2 = y^2\), we can determine that \(a = y\) because the square root of \(y^2\) is \(y\)

Step 2: Discover b

Next, consider \(2ab = -10y\). We know from step 1 that \(a = y\), so this becomes \(2*y*b = -10y\). Solve for \(b\) by isolating \(b\) on one side of the equation. Divide both sides of the equation by \(2*y\) to get \(b = -10y / 2*y\), which simplifies to \(b = -5\)

Step 3: Verify b

Lastly, make sure the \(b\) value obtained in step 2 corresponds with \(b^2\) in the equation. Here, \(b^2\) corresponds to \(25\). If \(b = -5\), then squaring \(b\) gives \(25\) which verifies our solution correct.

Step 4: Compose the binomial

In the last step, write down the original binomial square formula. Substituting the variables \(a\) and \(b\) with their new values, i.e, \(a = y\) and \(b = -5\), the binomial \(a+b\) becomes \(y + (-5)\) or \(y - 5\) and the whole expression will be of the form \((y - 5)^2\).

Key Concepts

Understanding Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and operators like addition, subtraction, multiplication, and division. These expressions can be as simple as a single number or variable, or they can be as complex as a combination of several terms. In our context, algebraic expressions help us describe the polynomial structure and understand its manipulation.

When dealing with algebraic expressions, keep these points in mind:

• Variables are symbols used to represent unknown values. In an expression, they often stand in for numbers whose values might change.
• An operator is a symbol that shows a mathematical operation, such as "+" for addition or "-" for subtraction.
• Constants are the numbers within an expression that have a fixed value.

The ultimate goal with these expressions is often to simplify, factor, or expand as needed, which makes understanding terms essential.

Introduction to Factoring

Factoring is the process of breaking down an algebraic expression into simpler pieces known as factors. This involves finding factors that, when multiplied together, give you the original expression. In our exercise, identifying whether an expression is the square of a binomial involves factoring.

To factor a quadratic expression like the one given, understand these concepts:

• A binomial is a type of algebraic expression with two terms, typically connected by a plus or minus sign.
• Factoring can reveal patterns like \(a + b\)^2, where this pattern highlights the expanded form of a squared binomial.
• By comparing the expanded quadratic term to a squared binomial formula, we verify if factoring reveals compatibility.

Recognizing these patterns aids in rewriting the expression to identify squares of binomials clearly.

Exploring Polynomials

Polynomials are expressions that consist of variables raised to whole number powers and their coefficients. Given our exercise, the expression \(y^2 - 10y + 25\) is a polynomial.

Here are key attributes of polynomials to understand better:

• Terms in a polynomial are parts of the expression separated by a plus or minus sign. Each term consists of a coefficient and a variable raised to an exponent.
• The degree of a polynomial is the highest power of the variable present in the expression. In our example, the degree is 2, making it a quadratic polynomial.
• Combining like terms and using methods like factoring can transform and simplify polynomials, sometimes revealing special identities or binomials.

Understanding these concepts is crucial as they lay the groundwork for solving polynomials, factoring them, and identifying patterns within them.