Problem 4 - Elementary Algebra

Question

Find the following products. $$ (3 a+b)^{2} $$

Step-by-Step Solution

Answer

Answer: The square of the binomial expression \((3a+b)^2\) is \(9a^2 + 6ab + b^2\).

Step 1: Identify the terms in the binomial expression

In our given expression, \((3a+b)^2\), we can identify the two terms as \(A = 3a\) and \(B = b\).

Step 2: Use the binomial square formula

Now, we can use the binomial square formula to expand the expression. The formula is \((A + B)^2 = A^2 + 2AB + B^2\). Substitute \(A = 3a\) and \(B = b\) into the formula.

Step 3: Find the square of each term and the product of both terms

Calculate \(A^2\), \(2AB\), and \(B^2\): \(A^2 = (3a)^2 = 9a^2\) \(2AB = 2(3a)(b) = 6ab\) \(B^2 = (b)^2 = b^2\)

Step 4: Combine the results

Now, combine these results based on the binomial square formula. So, \((3a+b)^2 = 9a^2 + 6ab + b^2\).

Key Concepts

Algebra

Algebra is a branch of mathematics that deals with symbols and the rules for manipulating these symbols. These symbols represent numbers and express operations and relationships among them. In the context of our exercise, algebra provides the tools necessary to expand and simplify polynomial expressions. Think of algebra as a language used to describe and solve problems involving unknown quantities. These unknowns are generally represented by letters, such as the letters "a" and "b" in our exercise. By applying algebraic rules, such as the distribution and commutative properties, we can simplify complex polynomial expressions like \((3a + b)^2\). This simplification helps find solutions to equations or to understand more about the relationship between the quantities involved.

Polynomial Expansion

The process of polynomial expansion involves rewriting a compact form like \((3a+b)^2\) into its expanded version, which is \(9a^2 + 6ab + b^2\). This is achieved using the binomial theorem or special formulas available in algebra.A polynomial is an expression consisting of variables, also known as indeterminates, and coefficients. It only involves addition, subtraction, multiplication, and positive whole number exponents of variables. To expand a binomial squared, \((A + B)^2\), we use the formula: \(A^2 + 2AB + B^2\). This formula tells us how to create each term in the expansion:

• Square the first term, \(A^2\).
• Multiply the first and second terms and double it, \(2AB\).
• Square the last term, \(B^2\).

Substitute \(A\) and \(B\) from the original expression to find each expanded polynomial term. This expansion process helps translate expressions into forms that are easier to work with.

Quadratic Expressions

Quadratic expressions are polynomials of degree two. They are crucial in both algebra and calculus and often appear in the form \(ax^2 + bx + c\). In our example, the quadratic expression \((3a + b)^2\) expands to \(9a^2 + 6ab + b^2\).Quadratic expressions have unique properties:

• They can plot as parabolas on a graph.
• They feature a leading term with a squared variable.
• They can be factored or expanded to simplify or solve equations.

Understanding quadratic expressions involves recognizing the symmetry and structure of parabolas. These expressions appear in numerous real-world scenarios, such as calculating areas or modeling projectile motions. The ability to expand or factor them as needed is fundamental in solving equations and performing polynomial operations effectively. Through practice and application, working with quadratic expressions becomes more intuitive, making resolution of problems involving them more straightforward.