Question

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

Short answer

2x (3x + 5)

Step-by-step solution

- Distribute and Expand

Start by expanding the given expression: \( 2x(2x + 5) + x^2 \times 2 \). Distribute the terms: \( 4x^2 + 10x + 2x^2 \).

- Combine Like Terms

Combine the like terms in the expression: \( 4x^2 + 2x^2 + 10x \). Simplify this to get: \( 6x^2 + 10x \).

- Factor Out the Greatest Common Factor (GCF)

Identify the greatest common factor (GCF) of the terms. Here, both terms share a common factor: \( 2x \). Factor out \( 2x \) from the expression: \( 2x (3x + 5) \).

Key concepts

Distributive Property

The distributive property is a useful tool in algebra that lets you multiply a sum by distributing, or spreading, the multiplication over each addend inside the parentheses. It means that if you have something of the form \( a(b + c) \), it can be expanded to \( ab + ac \). In our example, the expression is \( 2x(2x + 5) + x^2 \times 2 \). First, we distribute the \( 2x \) inside the parentheses: \( 2x \times 2x + 2x \times 5 \). This gives us \( 4x^2 + 10x \). We also multiply \( x^2 \times 2 \) to get \( 2x^2 \). Now, combining these results gives us \( 4x^2 + 10x + 2x^2 \). By using the distributive property, we can simplify complex expressions step by step.

Combining Like Terms

After distributing, the next step is to combine like terms. Like terms are terms that have the same variables raised to the same power. In our expression \( 4x^2 + 10x + 2x^2 \), the like terms are \( 4x^2 \) and \( 2x^2 \). We combine these by adding their coefficients: \( 4x^2 + 2x^2 = 6x^2 \). Now our simplified expression looks like this: \( 6x^2 + 10x \). Combining like terms makes the expression easier to manage and prepares it for further factoring or solving.

Greatest Common Factor

Factoring out the greatest common factor (GCF) is an important step in simplifying expressions. The GCF is the largest number and variable(s) that divide all terms in the expression. For \( 6x^2 + 10x \), we look for the GCF of both terms. Both terms share the number 2 and the variable x. The GCF is therefore \( 2x \). We factor out \( 2x \) from each term: \( 6x^2 + 10x \) becomes \( 2x (3x + 5) \). So, the factored form of our expression is \( 2x (3x + 5) \). Factoring out the GCF simplifies the expression and can help in solving equations or further factoring.