Question

Find the product. $$(3 x+5)(3 x+5)$$

Short answer

The product of the two binomials \((3x + 5)(3x + 5)\) is \(9x^2 + 30x + 25\).

Step-by-step solution

Identify the Binomial Terms

First identify the terms of the binomial. In this case, \(a = 3x\) and \(b = 5\).

Apply the Formula

Now, apply the perfect square binomial formula which is \((a+b)^2 = a^2 + 2ab + b^2\). Substituting \(a = 3x\) and \(b = 5\), the expression becomes \((3x + 5)^2 = (3x)^2 + 2*(3x)*(5) + (5)^2\).

Simplify the Terms

Simplify each term to get \(9x^2 + 30x + 25\).

Key concepts

Binomial Multiplication

To fully understand how to multiply binomials, it's crucial to get a grip on the pattern they commonly follow. A binomial is a polynomial with exactly two terms, such as 3x and 5 in the binomial expression (3x + 5). When we multiply two binomial expressions, especially when they are the same, this is called squaring a binomial. To multiply binomials, we use the distributive property, often referred to as the FOIL method, which stands for First, Outer, Inner, and Last terms.

Applying the FOIL method

In our exercise, when multiplying (3x + 5) by (3x + 5), we multiply the First terms (3x * 3x), the Outer terms (3x * 5), the Inner terms (5 * 3x), and the Last terms (5 * 5). However, because we are squaring a binomial, we can use a shortcut known as the perfect square binomial formula: ^2 = a^2 + 2ab + b^2. This formula simplifies the process and minimizes the potential for mistakes.

Algebraic Expressions

An algebraic expression is a combination of variables, numbers, and at least one arithmetic operation. In our example, (3x + 5) is an algebraic expression with '3x' and '5' as terms, 'x' as the variable, '3' and '5' as coefficients or constants respectively, and the '+' as the arithmetic operation. Understanding each part of an algebraic expression is foundational for manipulating and solving algebraic equations or inequalities.

Building Blocks of Algebraic Expressions

The exercise involves squaring the expression (3x + 5). This act of raising the expression to the power of two further illustrates how the individual terms interact algebraically when combined using operations such as multiplication.

Simplifying Algebraic Expressions

Simplifying an algebraic expression requires you to condense it into its simplest form, combining like terms, and performing arithmetic operations. For the expression (3x + 5)^2, simplification means executing the multiplication and then combining any like terms, which, in this case, are not present.

Steps in Simplification

Following the perfect square binomial formula, we first multiply the terms (3x * 3x), then the mixed terms (2 * 3x * 5), and lastly the constant terms (5 * 5), resulting in a simplified form of 9x^2 + 30x + 25. No further simplification is possible because each term is unique in regards to its variables and exponents. By simplifying algebraic expressions, we can efficiently evaluate them for different values of the variables or solve algebraic equations.