How to solve (3n ± 5)2n

Simplify the Problem

To solve $(3n \pm 5)2n$, we first need to understand the form and notation of the expression. It seems that we are handling two possible cases:

General Approach

For either case, we'll evaluate the expression by breaking it down step-by-step.

Case 1: $(3n + 5)2n$

We need to simplify $(3n + 5) \times 2n$:

Simplify further:

Case 2: $(3n - 5)2n$

Similarly, we evaluate $(3n - 5) \times 2n$:

Simplify further:

Final Result

To summarize the simplified forms of both cases:

In either scenario, the expression reduces to a quadratic polynomial. Understanding these steps will allow you to solve similar expressions effectively.

Simplification of polynomial expressions

Simplification of polynomial expressions involves combining like terms and performing arithmetic operations to rewrite the expression in its most reduced form.

Steps to Simplify Polynomial Expressions

1. Combine Like Terms: Like terms have the same variable raised to the same power. For example, $3x^2$ and $-5x^2$ are like terms.

2. Perform Arithmetic Operations: Carry out addition, subtraction, multiplication, and sometimes division of polynomials.

3. Rearrange Terms in Standard Form: The polynomial is usually written in descending order of the powers of the variables, i.e., from the highest degree to the lowest.

Example 1: Simplify $2x^2 + 3x - 5 + x^2 - 4x + 7$

1. Combine Like Terms:

   $$(2x^2 + x^2) + (3x - 4x) + (-5 + 7)$$

2. Perform the Arithmetic:

   $$3x^2 - x + 2$$

Example 2: Simplify $(x + 2)(x - 3)$

Use the distributive property, also known as FOIL (First, Outer, Inner, Last):

Example 3: Simplify $\frac{3x^2 + 6x}{3x}$

Divide each term in the numerator by the term in the denominator:

Important Formulas

For polynomials, some common formulas include:

1. Quadratic Formula:

   $$ax^2 + bx + c \rightarrow \text{standard form}$$

2. Product of Binomials:

   $$(a + b)(c + d) = ac + ad + bc + bd$$

Summary

The goal of simplification is to make the polynomial easier to work with, either by combining like terms or by performing algebraic manipulations, so that the expression is more compact and simplified.

Understanding the Distributive Property

The distributive property is a fundamental principle in algebra that describes how multiplication interacts with addition or subtraction. It can be stated as follows:

For any numbers $a$, $b$, and $c$:

Practical Application

The distributive property is particularly useful when you need to simplify expressions or solve equations. Here is an example:

Using the distributive property, you can break it down:

Which simplifies to:

Why It Matters

• Efficiency: It helps in solving complex problems more efficiently.
• Foundation for Advanced Mathematics: This property is foundational for more advanced topics like factoring polynomials.
• Mental Math: Enhances mental math skills by simplifying calculations.

Another Example with Variables

Consider an example with variables:

The distributive property ensures that multiplication distributes over addition, maintaining the equality.

Conclusion

The distributive property is a powerful tool in algebra that simplifies expressions and enhances problem-solving skills. Understanding and applying this property is essential for success in mathematics.