Question

Expressions that occur in calculus are given. Factor each expression completely. $$ 5(2 x+1)^{2}+(5 x-6) \cdot 2(2 x+1) \cdot 2 $$

Short answer

The completely factored form is \( (2x + 1) (30x - 19)\).

Step-by-step solution

Identify the common factor

First, observe that the expression consists of two terms: 1. \(5(2x+1)^2\) 2. \((5x-6) \times 2(2x+1) \times 2\). Notice that both terms contain the factor \((2x+1)\).

Factor out the common term

Factor out \((2x+1)\): \(5(2x+1)^2+(5x-6)\times 2(2x+1)\times 2 = (2x+1)\times[5(2x+1)+(5x-6)\times 2\times 2]\).

Simplify the simplified form

Focus on the simplified form, particularly what's in the brackets: \(5(2x+1)+(5x-6)\times 4\). We simplify it further by expanding: \(5(2x+1) + (5x-6) \times 4 = 10x + 5 + 20x - 24\).

Combine like terms

Combine the terms inside the brackets: \(10x + 20x +5 - 24 = 30x - 19\).

Write the final factored form

Merge the simplified terms back into the original form: \( (2x + 1) (30x - 19)\). This is the completely factored form of the original expression.

Key concepts

Factoring

Factoring is the process of breaking down an expression into products of simpler expressions. In algebra, this often involves finding two or more expressions that multiply together to form the original expression.
This process is especially important in solving equations and simplifying expressions because it reveals the structure of a polynomial.
For example, in the given problem, we started with the expression: \[ 5(2 x+1)^{2}+(5 x-6) \times 2(2 x+1) \times 2 \].
Factoring involves looking for common factors (which we will discuss next) and then using distributive properties to simplify the expression.
Ultimately, the goal is to express the polynomial in a more manageable form, such as \[ (2 x + 1) (30x - 19) \].

Common Factors

A common factor is a term that is present in each part (or term) of the expression. Identifying common factors is the key first step in the factoring process.
In the example problem, we identified \[ (2x+1) \] as a common factor shared by both terms \[ 5(2x+1)^2 \] and \[ (5x-6) \times 2(2x+1) \times 2 \].
If you see a term that appears in all parts of an expression, it can often be factored out to simplify the entire expression.
For instance, once we factor out \[ (2x+1) \], the expression looks less complex, becoming \[ (2x+1) [5(2x+1) + (5x-6) \times 4] \].
This step reduces the original polynomial and makes the next steps easier, such as combining like terms.

Polynomial Simplification

Polynomial simplification involves combining like terms and performing arithmetic operations to make an expression as concise as possible.
In our exercise, after factoring out the common factor, we had in the brackets: \[ 5(2x+1) + (5x-6) \times 4 \].
We then expanded and combined like terms:
\[ 5(2x+1) + (5x-6) \times 4 = 10x + 5 + 20x - 24 \].
Next, we combined the \[x \]-terms and the constant terms resulting in \[10x + 20x + 5 - 24 = 30x - 19 \].
This simplification process is pivotal in obtaining a cleaner, more understandable expression, which is crucial for further mathematical computations.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operation symbols. They are fundamental in algebra and come in various forms, like monomials, binomials, and polynomials.
For example, \[ 5(2x + 1)^{2} + (5x - 6) \times 2(2x +1) \times 2 \] is a polynomial.
To work effectively with algebraic expressions, understanding operations such as addition, multiplication, and factoring is essential.
The given expression was transformed step by step until we simplified and factored it completely into \[ (2x + 1) (30x - 19) \].
This illustrates how recognizing the structure of algebraic expressions and applying suitable operations can greatly simplify and clarify your work.