Problem 49 - A Graphical Approach to College Algebra

Question

Find each product. $$[(3 a+b)-1]^{2}$$

Step-by-Step Solution

Answer

The expanded expression is \(9a^2 + 6ab + b^2 - 6a - 2b + 1\).

Step 1: Identify the Expression

The expression given is \([(3a + b) - 1]^2\). We need to find the product by expanding this squared term.

Step 2: Apply the Square of a Binomial Formula

Recall that the square of a binomial \((x-y)^2\) is expanded using the formula: \[(x-y)^2 = x^2 - 2xy + y^2\].In our case, identify \(x = (3a + b)\) and \(y = 1\).

Step 3: Substitute into the Formula

Substitute \(x = (3a + b)\) and \(y = 1\) into the binomial square formula:\[((3a + b) - 1)^2 = (3a + b)^2 - 2(3a + b) imes 1 + 1^2\].

Step 4: Expand the Expression

First, expand \((3a + b)^2\) using the expansion:\((3a + b)^2 = (3a)^2 + 2(3a)(b) + b^2\) which results in:\(= 9a^2 + 6ab + b^2\).Now substitute back into the main expression:\((3a + b)^2 - 2(3a + b) + 1\).

Step 5: Distribute and Simplify

Now, simplify \(-2(3a + b)\):\(-2(3a + b) = -6a - 2b\).Combining all parts:\(9a^2 + 6ab + b^2 - 6a - 2b + 1\).

Step 6: Final Simplification

Combine like terms (if any), although here there are none to combine between the parts.So, the expanded form is:\[9a^2 + 6ab + b^2 - 6a - 2b + 1\].

Key Concepts

Algebra

Algebra is a crucial part of mathematics that helps us understand relationships between variables and constants. It's about finding the unknowns by manipulating equations and expressions.
In our exercise, we are dealing with an algebraic expression that involves variables like \(a\) and \(b\). This expression needs to be expanded to find its product when squared.

Key concepts in algebra include:

• Variables : Symbols that represent numbers or values, like \(a\) and \(b\) here.
• Constants : Fixed values that do not change, such as the number 1 in the expression.
• Operations : Addition, subtraction, multiplication, and division used to solve the expressions.

In this problem, using algebra helps us work through the process of binomial expansion, transforming a complex expression into something more manageable.

Polynomial Expressions

Polynomial expressions consist of variables, coefficients, and exponents linked by addition or subtraction. They're like mathematical sentences describing a relationship among numbers.
In this exercise, our expression \(((3a+b) - 1)^2\) includes a binomial (a two-term polynomial).

Let's break this down:

• Binomial : A type of polynomial with exactly two terms, such as \((3a + b)\).
• Square of a Binomial : Using the formula \( (x - y)^2 = x^2 - 2xy + y^2 \) to expand and simplify it.

In the expression \(((3a + b) - 1)^2\), we first identify \(x\) and \(y\), then apply the formula to expand it into a polynomial expression. This process turns it from a neat, concise square into a long expanded form that is easier to work with.

Mathematics Education

Mathematics education involves mastering concepts through practice and understanding. With algebra and polynomial expressions, students learn to handle complex problems in manageable steps.
In this exercise, breaking down problems like \( [(3a+b) - 1]^2 \) is an example of building math skills through structured steps.

Here's how the educational process unfolds:

• Identify : Recognizing the parts of the expression and how they fit together.
• Apply Formulas : Using known formulas like the square of a binomial to simplify the problem.
• Simplify : Combining the steps into a final, understandable result.

The exercise helps students see how algebraic manipulation and understanding can change an equation's form, making it a vital part of a comprehensive mathematics education. This ability not only helps in solving the current problem but builds the skills for tackling more complex ones in the future.