Problem 13 - Algebra 2

Question

Simplify. $$ (2 x-1)\left(x^{2}-4 x+4\right) $$

Step-by-Step Solution

Answer

The simplified expression is \(2x^3 - 9x^2 + 12x - 4\).

Step 1: Distribute the First Term

To simplify the expression \((2x - 1)(x^2 - 4x + 4)\), we first distribute the first term in the first parenthesis across each term in the second parenthesis. This involves multiplying \(2x\) by each term in \(x^2 - 4x + 4\):\[2x \cdot x^2 = 2x^3\]\[2x \cdot (-4x) = -8x^2\]\[2x \cdot 4 = 8x\]

Step 2: Distribute the Second Term

Next, distribute the second term of the first parenthesis, \(-1\), across each term in the second parenthesis. Multiply \(-1\) by each term in \(x^2 - 4x + 4\):\[-1 \cdot x^2 = -x^2\]\[-1 \cdot (-4x) = 4x\]\[-1 \cdot 4 = -4\]

Step 3: Combine Like Terms

Now that we have distributed each term, combine the like terms from all the products:- Combining terms for \(x^2\): \(-8x^2\) and \(-x^2\) gives \(-9x^2\).- Combining terms for \(x\): \(8x\) and \(4x\) gives \(12x\).This results in the simplified expression:\[2x^3 - 9x^2 + 12x - 4\]

Step 4: Final Expression

No further simplification of the expression \(2x^3 - 9x^2 + 12x - 4\) is possible, as all like terms have been combined.

Key Concepts

The Distributive Property

The distributive property is a fundamental concept in algebra. It allows us to multiply a single term by each of the terms in a parenthesis. In the equation \((2x - 1)(x^2 - 4x + 4)\), you use this property to simplify the expression step by step.

Here's how it works:

• Take the first term from the first parenthesis, which is \(2x\), and multiply it by each term in the second parenthesis \((x^2 - 4x + 4)\).
• Similarly, take the second term from the first parenthesis, which is \(-1\), and distribute it across each term in the second parenthesis as well.

This process ensures each term in one polynomial gets distributed across every term in the other, breaking complex expressions into simpler, manageable parts. The key here is to methodically multiply and add (or subtract) so that no term is left out. This step sets the foundation for combining like terms later.

Combining Like Terms

Combining like terms is an essential skill in simplifying expressions. A 'like term' refers to parts of the expression that have the same variable raised to the same power. Once all terms are distributed, it's time to look for these similar terms.

In our example, we distribute terms and get several expressions that need combining:

• \(-8x^2\) and \(-x^2\) both have the variable raised to the second power, making them like terms. Combine them by adding their coefficients: \(-8 + (-1) = -9\).
• \(8x\) and \(4x\) are also like terms as they both have the variable raised to the first power. Combine their coefficients: \(8 + 4 = 12\).

This process of combining like terms helps to consolidate the expression into fewer, simpler terms. Each step reduces clutter and brings clarity, ultimately leading you to the simplest form of the expression, which in our case ends up as \(2x^3 - 9x^2 + 12x - 4\).

Polynomial Multiplication

Polynomial multiplication can seem daunting at first, but it's just an extension of the distributive property. The principle here is to ensure every term in one polynomial multiplies with every term in another, accounting for all combinations.

Consider our expression \((2x - 1)(x^2 - 4x + 4)\):

• Multiply each term in the first polynomial by each term in the second. This results in individual products like \(2x \times x^2 = 2x^3\) and \(-1 \times 4 = -4\).
• It helps to organize these products clearly to make the next step, combining like terms, more manageable.
• Always check that each product has been calculated correctly to avoid mistakes.

So, polynomial multiplication is about thoroughness and organization. By following these steps systematically, complex expressions become solvable, and you'll arrive at the simplified form, ensuring that no components are overlooked in the process.