Question

Expressions that occur in calculus are given. Factor each expression completely. $$ 3(4 x+5)^{2} \cdot 4(5 x+1)^{2}+(4 x+5)^{3} \cdot 2(5 x+1) \cdot 5 $$

Short answer

\[ (4x + 5)^2 (5x + 1) (100x + 62) \]

Step-by-step solution

- Identify Common Factors

Examine the given expression: \[ 3(4x+5)^2 \times 4(5x+1)^2 + (4x+5)^3 \times 2(5x+1) \times 5 \].Notice that \(4x+5\) and \(5x+1\) are common factors in both terms. Specifically, factor out \( (4x+5)^2 \times (5x+1) \).

- Factor Out Common Terms

Factor out \( (4x+5)^2 \times (5x+1) \) from both terms:\[ (4x+5)^2 \times (5x+1) \times [12(5x+1) + 10(4x+5)] \].

- Simplify Inside the Bracket

Simplify the expression inside the bracket:\[ 12(5x + 1) + 10(4x + 5) \]. Distribute the constants:\[ 60x + 12 + 40x + 50 \]. Combine like terms: \[ 100x + 62 \].

- Write the Final Factored Form

The fully factored form of the expression is:\[ (4x + 5)^2 (5x + 1) (100x + 62) \].

Key concepts

Identifying Common Factors

To begin factoring any algebraic expression, it's important to first identify common factors. This means finding terms that appear in each part of the expression.

In the exercise given, we start with the expression: \( 3(4x+5)^2 \times 4(5x+1)^2 + (4x+5)^3 \times 2(5x+1) \times 5 \)

Upon examining it closely, you'll notice that both parts of the expression have \( (4x+5) \) and \( (5x+1) \) as factors.

To make the next steps easier, we factor out the common parts first.

This will simplify our expression and make it more manageable to solve. Identifying common factors is like finding the greatest common divisor in a set of numbers; it helps streamline the problem-solving process.

Distributive Property

Understanding the distributive property is crucial when factoring algebraic expressions.

The distributive property states that for any numbers \(a, b, \text{and} \, c, a(b + c) = ab + ac \). This means you distribute the term outside the parentheses to each term inside the parentheses.

In our example, after identifying the common factors, we rewrote the expression as: \( (4x+5)^2 \times (5x+1) \times [12(5x+1) + 10(4x+5)] \).

We used the distributive property inside the bracket to expand the terms:

\[ 12(5x + 1) + 10(4x + 5) = 60x + 12 + 40x + 50 \].

By breaking it down and using the distributive property, we made the problem simpler to handle.

Combining Like Terms

Combining like terms is the process of simplifying an algebraic expression by merging terms that have the same variables and exponents.

After distributing, we had: \[ 60x + 12 + 40x + 50 \].

Next, we combine like terms. Here, \(60x \) and \(40x \) are like terms because they both contain the variable \( x \). Similarly, \(12 \) and \(50 \) are constant terms and can be combined:

\[ 60x + 40x = 100x \]
\[ 12 + 50 = 62 \].

So, the simplified expression inside the bracket becomes: \[ 100x + 62 \].

Combining like terms is vital in simplifying expressions to their most reduced form, making further calculations easier.