Problem 25 - College Algebra with Corequisite Support

Question

For the following exercises, expand the binomial \((3 y-7)^{2}\)

Step-by-Step Solution

Answer

The expanded form is \(9y^2 - 42y + 49\).

Step 1: Identify the Binomial Formula

To expand the binomial \((3y - 7)^2\), use the formula for the square of a binomial: \((a - b)^2 = a^2 - 2ab + b^2\). Here, \(a = 3y\) and \(b = 7\).

Step 2: Substitute Values into the Formula

Substitute \(a = 3y\) and \(b = 7\) into the binomial formula. This gives us: \((3y)^2 - 2(3y)(7) + 7^2\).

Step 3: Calculate Each Term

Calculate each term separately: 1. \((3y)^2 = 9y^2\)2. \(-2(3y)(7) = -42y\)3. \(7^2 = 49\).

Step 4: Combine the Terms

Combine the calculated terms to get the expanded form: \(9y^2 - 42y + 49\).

Key Concepts

Algebra

Algebra is a fundamental branch of mathematics that uses symbols, letters, and numbers to express relationships and solve equations. It allows us to create mathematical models of real-world situations and solve problems systematically. In the context of expanding the binomial \((3y-7)^2\), algebra helps us follow a series of logical steps to determine the expanded form.

Using algebraic techniques, you break down complex expressions into simpler terms introduced as variables or constants.

• Variables, like \(y\) in the expression, symbolize quantities that can change.
• Constants, like numbers 3 and 7, stand for fixed values.

To expand a binomial using algebra, you leverage formulas like the square of a binomial formula. This formula assists in systematically transforming \((3y - 7)^2\) into a more digestible expression, accommodating for our understanding of changes in quantities.

Polynomials

Polynomials are expressions made up of variables and coefficients, involving operations like addition, subtraction, and non-negative integer exponents of variables. They are an essential part of algebra and come in various forms, ranging from simple monomials to complex multi-term expressions.

In the context of our expression \((3y-7)^2\), expanding it creates a polynomial. These are expressed as the sum of its terms:

• First Term: \(9y^2\)
• Second Term: \(-42y\)
• Third Term: \(49\)

Each term is a distinct part of the polynomial, identified by its coefficient (like 9, -42, 49) and variable part (like \(y^2, y\)). When you combine these terms, you're working with a polynomial in its expanded form, which helps understand the structure and behavior of algebraic equations.

Quadratic Expressions

Quadratic expressions are a type of polynomial, specifically, they are polynomials of degree 2. This means that the highest power of the variable in the expression is 2. These types of expressions often take the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are coefficients.

The expansion process of \((3y-7)^2\) results in a quadratic expression: \(9y^2 - 42y + 49\). Quadratics are key in modeling scenarios where the relationships between variables follow a parabolic curve—like projectile motion or optimizing areas.

• The term \(9y^2\) signifies the square term, which determines the parabola's direction (up or down).
• \(-42y\) is the linear term, influencing the slope of the curve at different points.
• Finally, \(49\) is the constant term, affecting the location of the curve along the y-axis.

Graphing quadratic expressions or solving them helps students develop insights into diverse mathematical and practical problems.