Problem 69 - Elementary and Intermediate Algebra

Question

Multiply. See Example 8. $$ 4 x(2 x+1)(x-2) $$

Step-by-Step Solution

Answer

The product is \(8x^3 - 12x^2 - 8x\).

Step 1: Distribute the First Set of Terms

First, distribute the 4x across the terms inside the first set of parentheses (2x+1). This step involves multiplying each term inside the parentheses by 4x:\[ 4x \cdot 2x = 8x^2 \]\[ 4x \cdot 1 = 4x \]So, the expression becomes (8x^2 + 4x).

Step 2: Expand the Expression

Next, multiply the result from Step 1 by the terms in the second set of parentheses (x-2). This requires distributing each term from Step 1 with each term in the new parentheses:\[ (8x^2 + 4x) \cdot (x - 2) \]This can be broken into two major multiplications: one for each term in the first expression.

Step 3: Distribute on the First Term

Multiply the first term (8x^2) by each term inside the second parentheses (x-2):\[ 8x^2 \cdot x = 8x^3 \]\[ 8x^2 \cdot (-2) = -16x^2 \]

Step 4: Distribute on the Second Term

Multiply the second term (4x) by each term in the parentheses (x-2):\[ 4x \cdot x = 4x^2 \]\[ 4x \cdot (-2) = -8x \]

Step 5: Combine Like Terms

Add all the products from Steps 3 and 4 together to form a single polynomial:\[ 8x^3 - 16x^2 + 4x^2 - 8x \]Combine the like terms (the \(x^2\) terms):\[ 8x^3 - 12x^2 - 8x \]

Step 6: Final Expression

The final expression after all multiplications and combinations of like terms is:\[ 8x^3 - 12x^2 - 8x \]

Key Concepts

Distributive Property

The distributive property is a fundamental rule in algebra that allows us to simplify expressions. It states that for any three numbers or variables, the product of a term outside the parentheses and each term inside it must be calculated separately. This is expressed as: \( a(b+c) = ab + ac \).
In algebraic expressions like \( 4x(2x+1) \), the distributive property guides us to distribute \( 4x \) across \( 2x \) and \( 1 \). We multiply each term separately to obtain \( 8x^2 \) and \( 4x \). This step is crucial as it breaks down complex expressions into simpler parts.

• Apply the property when you see parentheses preceded by a term.
• Multiply sequentially and systematically to avoid mistakes.
• Keep track of signs, especially with subtraction.

Mastering the distributive property is essential for more complicated algebraic manipulations.

Combine Like Terms

Combining like terms is an essential skill in simplifying polynomials. "Like terms" are terms that have the same variable raised to the same power. This means their coefficients can be added or subtracted.
For example, in an expression such as \( 8x^3 - 16x^2 + 4x^2 - 8x \), we identify like terms: \( -16x^2 \) and \( 4x^2 \) can be combined:
- Collect like terms - Add or subtract their coefficients - Keep the variable and exponent the same
By combining \( -16x^2 + 4x^2 \), we simplify the expression to \( -12x^2 \). This simplifies the polynomial and makes it easier to understand and work with. Keeping these terms organized is crucial as it affects the outcome of the expression.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations. They form the basis for algebra and give us a way to model real-world situations or solve problems. Each expression can include coefficients, constants, variables, and operators like addition or multiplication.
For instance, expressions like \( 4x(2x+1)(x-2) \) involve several steps to simplify, including distribution and combining like terms. Understanding how to navigate these expressions helps with factorization, solving equations, and graphing.

• Identify each component: coefficients, variables, constants
• Use operations strategically to simplify or expand the expression
• Practice with a variety of expressions to enhance familiarity

Recognizing and working with algebraic expressions is a foundational skill that builds towards more advanced mathematical problem-solving.