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When you encounter an expression like \(x(x+2)(2x-1)\), you need to expand it to simplify. Polynomial expansion involves multiplying terms to remove parentheses and rewrite the expression in a simpler form. This step might seem complex at first, but it's merely about applying multiplication rules across the different terms inside parentheses.

To expand, you start by distributing one term across the terms in the parentheses. Here, \(x\) is distributed over \((x+2)\), leading to \(x^2 + 2x\). Then, you take this new expression \((x^2 + 2x)\) and distribute it over the next binomial \((2x-1)\). This is where polynomial expansion becomes crucial as it sets the stage for combining and simplifying the expression later on.

After you expand the polynomial, the next step is to combine like terms. Like terms are terms in an expression that have the same variable raised to the same power. These can be added together to simplify the expression.

For example, once you multiply everything out from our polynomial \((x^2 + 2x)(2x-1)\), you get a mix of terms: \(2x^3 - x^2 + 4x^2 - 2x\). Notice the \(x^2\) terms: \(-x^2\) and \(4x^2\). These are like terms because they have the same variable \(x\) and the same exponent \(2\).

• Combine \(-x^2 + 4x^2\) to get \(3x^2\).

After combining, your polynomial is much simpler: \(2x^3 + 3x^2 - 2x\). This makes it easier to understand and use, whether you're plugging in values or using it in further calculations.

The distributive property is a principle of multiplication that allows you to deal with expressions inside parentheses. Mathematically, it's written as \(a(b+c) = ab + ac\). Basically, it tells us how to multiply expressions by breaking them down into simpler steps.

Using the distributive property, you multiply each term inside the parentheses by the term outside. This method helps in expanding polynomials, like our initial expression, by breaking down the multiplication into manageable parts.

• First, \(x(x+2)\) distributes to \(x^2 + 2x\).
• Then, expand \((x^2 + 2x)(2x-1)\) by distributing \(x^2\) and \(2x\) separately across \(2x - 1\).

By applying the distributive property, you ensure that each term is correctly multiplied and added, leading to the expanded and simplified expression \(2x^3 + 3x^2 - 2x\). This property is key when working with algebraic expressions to prevent mistakes and ensure accuracy.