Problem 43 - A Graphical Approach to College Algebra

Question

Find each product. $$\left(5 r+3 t^{2}\right)^{2}$$

Step-by-Step Solution

Answer

The expanded product is \(25r^2 + 30rt^2 + 9t^4\).

Step 1: Recognize the Formula

The expression \((5r + 3t^2)^2\) is a binomial raised to the power of 2. This can be expanded using the binomial square formula: \((a + b)^2 = a^2 + 2ab + b^2\), where \(a = 5r\) and \(b = 3t^2\).

Step 2: Calculate \(a^2\)

Square the first term: \((5r)^2 = 25r^2\).

Step 3: Calculate \(b^2\)

Square the second term: \((3t^2)^2 = 9t^4\).

Step 4: Calculate \(2ab\)

Calculate two times the product of the first and second terms: \(2 \cdot 5r \cdot 3t^2 = 30rt^2\).

Step 5: Combine All Parts

Use the expanded terms to express the binomial: \(25r^2 + 30rt^2 + 9t^4\). This is the expanded form of the expression.

Key Concepts

Polynomials

A polynomial is an expression made up of variables, coefficients, and exponents, all combined using addition, subtraction, and multiplication. Polynomials can have one or more terms. Each term in a polynomial consists of:

• A variable (such as \(r\) or \(t\) in our example) that can take different values.
• An exponent that denotes the power to which the variable is raised (2 in \(r^2\)).
• A coefficient, which is a constant multiplied by the variable (like 5 in \(5r\)).

Polynomials are classified based on their degree and the number of terms. The degree of a polynomial is determined by the highest exponent it contains. For instance, in the polynomial \(25r^2 + 30rt^2 + 9t^4\), the degree is 4, contributed by the term \(9t^4\). Understanding polynomials is key in algebra as they form the foundation for many mathematical concepts.

Algebraic Expressions

Algebraic expressions include numbers, variables, and operations, all arranged to describe a specific quantity or formula. In simple terms, they represent a statement of equality or an expression of calculation using algebraic symbols.

An example of such an expression is \((5r + 3t^2)^2\), which involves both addition and exponentiation. The purpose of algebraic expressions is to succinctly convey mathematical ideas, relationships, or structures.

• Terms like \(5r\) and \(3t^2\) are algebraic terms that can be combined or simplified depending on the operations involved.
• The expression can be evaluated for particular values of the variables \(r\) and \(t\), providing numeric results.
• They can represent equations or inequalities and are pivotal in forming models in science and engineering.

Understanding how to manipulate and simplify algebraic expressions is a crucial skill in solving complex equations.

Exponentiation

Exponentiation is a mathematical operation involving two numbers: the base and the exponent. It simplifies repeated multiplication of the same number or expression.

For example, in the expression \((5r + 3t^2)^2\), the 2 is the exponent, applied to the base \((5r + 3t^2)\). This means to multiply \((5r + 3t^2)\) by itself:

• This operation uses the power of 2, simplifying repeated multiplication by indicating it concisely.
• The binomial expansion using exponentiation is crucial, applying the formula \((a + b)^2 = a^2 + 2ab + b^2\), an example of exponentiation in action.
• Exponentiation in algebra is just one way it shows up; it's also prevalent in areas like calculus, physics, and engineering.

The concept of exponentiation is important in algebraic operations as it plays a significant role in calculations and can simplify otherwise complex mathematical expressions.