Question

Explain how to use the distributive property to find the product. $$ (2 x-3)(x+4) $$

Short answer

The simplified form of (2x - 3)(x + 4) using the distributive property is \(2x^2 + 5x - 12\).

Step-by-step solution

Understanding the problem and the distributive property

We are asked to simplify the expression (2x - 3)(x + 4) using the distributive property. The distributive property is a(b + c) = ab + ac, meaning you multiply each term inside the bracket by the term outside.

Apply the distributive property

First, distribute 2x to the terms inside the second brackets: (2x*x + 2x*4). You will get an equation \(2x^2 + 8x\). Secondly, distribute -3 to the terms inside the second brackets: (-3*x -3*4). This will give you \(-3x - 12\).

Combine like terms

Now, combine the result obtained in the previous step: \(2x^2 + 8x - 3x - 12\). We can combine the like term 8x and -3x to get 5x. So, the expression simplifies to \(2x^2 + 5x - 12\).

Key concepts

Simplifying Expressions

In algebra, simplifying expressions means transforming them into their most basic form without changing the value or the properties that the expressions hold. It's all about making equations less complex and more understandable. Think of it like packing for a trip – you want everything you need to fit in as few bags as possible.

Take the expression \( (2x - 3)(x + 4) \) for instance. When simplifying, the order of operations must be followed, and in this case, the distributive property becomes our main tool. Here's a breakdown: firstly, \(2x\times x + 2x\times 4\) is much simpler than looking at the original brackets. Immediately you can see that this will become \(2x^2 + 8x\). Next, \( -3 \times x - 3 \times 4\) simplifies to \( -3x - 12\). This uncomplicated approach is key to understanding complex algebraic concepts.

Remember, the aim isn't just to get an answer but to rewrite the equation in a way that reveals its structure clearly and makes it easier to use in subsequent calculations.

Combining Like Terms

When you have an expression laid out before you, spotting 'like terms' is like recognizing members of the same family in a crowd. They share common features – in this case, the same variables raised to the same power. In the expression \( 2x^2 + 8x - 3x - 12 \), the like terms are \( 8x \) and \( -3x \).

Merging them requires simple arithmetic. Think of \( +8x\) and \( -3x \) as directions on a number line. Moving 8 units to the right and then 3 units to the left brings you to a new position, \( +5x\). Combining these terms brings us the simplified expression \( 2x^2 + 5x - 12 \). This step is crucial in simplification because it gets rid of unnecessary clutter in the expression and paves the way for clearer computation or further algebraic manipulation.

Multiplying Polynomials

If simplifying expressions is packing for a trip, then multiplying polynomials is like planning your travel itinerary – you need to consider every possible route to get to your destination. Multiplying polynomials involves taking each term in one polynomial and multiplying it by every term in another polynomial.

For the expression \( (2x - 3)(x + 4) \), the distributive property is our guidebook. We systematically take each term from the first polynomial, starting with \(2x\) and then \( -3\), and multiply it by both terms in the second polynomial, \(x\) and \( +4\). By doing so, you map out all possible products: \(2x^2\), \(8x\), \( -3x\), and \( -12\). It's essential to be methodical here, much like ensuring you don't miss a flight connection – if you do, the entire trip (or in our case, the final expression) can be effected. Through multiplication, we expand and rewrite the expression into a form that can be further simplified by combining like terms, getting us one step closer to our algebraic destination.