Problem 41 - Elementary Algebra

Question

Simplify. $$ 2 x 3 x-1-13 x+1+2(x-1)(3 x-1)(3 x+1) $$

Step-by-Step Solution

Answer

The simplified expression is \(20x^3 - 18x^2 - 11x - 2\).

Step 1: Apply the Distributive Property

Begin by expanding the term \(2(x-1)(3x-1)(3x+1)\). This term can be seen as the product of \((3x-1)(3x+1)\), which is a difference of squares. The difference of squares formula states that \((a-b)(a+b) = a^2 - b^2\). Here, \(a = 3x\) and \(b = 1\), giving us: \[(3x)^2 - 1^2 = 9x^2 - 1\]Therefore, \((3x-1)(3x+1) = 9x^2 - 1\), and the expression becomes:\[2(x-1)(9x^2 - 1)\]

Step 2: Distribute Terms in Expansion

Now, distribute the term \(2(x-1)\) to \(9x^2 - 1\):- First, distribute 2 to each element: \[ 2(x-1)(9x^2) - 2(x-1)(1) \]- Apply another distributive property: \[2x(9x^2) - 2(9x^2) + 2x(1) - 2(1)\]- Simplify this: \[18x^3 - 18x^2 + 2x - 2\]

Step 3: Simplify the Original Expression

Rewrite the original expression by incorporating what you calculated:\[2x^3 - 1 - 13x + 1 + 18x^3 - 18x^2 + 2x - 2\]Now, simplify by combining like terms. Gather terms in order of decreasing degrees across all terms:

Step 4: Combine Like Terms

1. Combine the \(x^3\) terms: \[2x^3 + 18x^3 = 20x^3\]2. Combine the \(x^2\) terms: \[0 - 18x^2 = -18x^2\]3. Combine the \(x\) terms: \[-13x + 2x = -11x\]4. Combine constant terms: \[-1 + 1 - 2 = -2\]Thus, the expression simplifies to:\[20x^3 - 18x^2 - 11x - 2\]

Step 5: Final Simplified Expression

The expression, fully simplified, is:\[20x^3 - 18x^2 - 11x - 2\]

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that helps simplify expressions. It allows you to multiply a single term across other terms inside a set of parentheses. The basic formula for the distributive property is:

• \( a(b + c) = ab + ac \)

This means you distribute the outside term, \(a\), to each term inside the parentheses, \(b\) and \(c\).
In the original problem, we apply the distributive property several times. Initially, it’s used to expand \(2(x-1)(9x^2 - 1)\). You distribute \(2(x-1)\) to both \(9x^2\) and \(-1\).
Firstly, distribute \(2(x-1)\) with \(9x^2\) and then \(2(x-1)\) with \(-1\), giving us:

• \(2x(9x^2) - 2(9x^2) + 2x(1) - 2(1)\)

Simplifying each part leads to the new expression:

• \(18x^3 - 18x^2 + 2x - 2\)

The distributive property is a powerful tool that allows for breaking down complex expressions into simpler ones.

Difference of Squares

The difference of squares is another handy algebraic tool for simplifying expressions. It is the product of two conjugate binomials,

• \((a - b)(a + b) = a^2 - b^2\)

This means you take the square of the first term, \(a\), and subtract the square of the second term, \(b\).
In the given exercise, the terms

• \((3x-1)(3x+1)\)

fit this pattern. By applying the difference of squares, we get

• \((3x)^2 - 1^2 = 9x^2 - 1\)

This reduced form

• \(9x^2 - 1\)

makes it easier to combine with other terms in the expression when distributing. Understanding the difference of squares helps simplify polynomials quickly, turning daunting expressions into manageable pieces.

Like Terms

Combining like terms is an essential skill in simplifying algebraic expressions. Like terms are terms that have the same variable raised to the same power. You can only combine coefficients of these like terms.
In the exercise, once the terms are expanded and simplified, it's time to combine like terms:

• Combine the \(x^3\) terms: \(2x^3 + 18x^3 = 20x^3\)
• Combine the \(x^2\) terms: \(0 - 18x^2 = -18x^2\)
• Combine the \(x\) terms: \(-13x + 2x = -11x\)
• Finally, combine the constant terms: \(-1 + 1 - 2 = -2\)

The simplified expression is

• \(20x^3 - 18x^2 - 11x - 2\)

Recognizing and combining like terms is essential to arriving at the simplest form of an expression, ensuring your answer is clean and concise.