Problem 28 - A Graphical Approach to College Algebra

Question

Find each product. $$(4 r-1)(7 r+2)$$

Step-by-Step Solution

Answer

The product is \(28r^2 + r - 2\).

Step 1: Apply the Distributive Property

Use the distributive property to expand the expression by multiplying each term in the first parenthesis by each term in the second parenthesis: \((4r - 1)(7r + 2) = 4r \cdot 7r + 4r \cdot 2 - 1 \cdot 7r - 1 \cdot 2\).

Step 2: Multiply Each Pair of Terms

Perform the multiplication for each pair of terms:1. \(4r \cdot 7r = 28r^2\) 2. \(4r \cdot 2 = 8r\) 3. \(-1 \cdot 7r = -7r\) 4. \(-1 \cdot 2 = -2\)

Step 3: Combine Like Terms

Combine the terms by adding or subtracting them together:\(28r^2 + 8r - 7r - 2\). Combine \(8r\) and \(-7r\) to get \(r\):\(28r^2 + r - 2\).

Step 4: Write Final Expression

Write the final expanded expression after combining like terms:The product \((4r - 1)(7r + 2)\) is \(28r^2 + r - 2\).

Key Concepts

Distributive Property

The distributive property is a fundamental principle in algebra that lets you multiply a single term by two or more terms inside a parenthesis. When you apply it, you ensure that all terms are properly accounted for during multiplication.
Imagine you have the expression

• \((a + b) \cdot c\),

according to the distributive property, this can be expanded to

• \(a \cdot c + b \cdot c\).

Using this approach on

• \((4r - 1)(7r + 2)\),

we distribute each term in the first set of parentheses by each term in the second one:

• \(4r \cdot 7r\),
• \(4r \cdot 2\),
• \(-1 \cdot 7r\),
• \(-1 \cdot 2\).

This process helps break down complex expressions into simpler components for easier computation.

Combining Like Terms

After you've applied the distributive property, the next step is to simplify the expression by combining like terms. Like terms are terms that have the same variable raised to the same power.
For instance, in the expression

• \(28r^2 + 8r - 7r - 2\),

the like terms are

• \(8r\) and \(-7r\).

These can be combined by performing the addition or subtraction:

• \(8r - 7r = r\).

Thus, the expression simplifies to

• \(28r^2 + r - 2\).

Combining like terms is crucial as it reduces the expression to its simplest form, making it easier to interpret or solve.

Polynomial Expansion

Polynomial expansion involves taking an expression involving multiple terms and expanding it into a sum or difference of terms. It often uses both the distributive property and the combining of like terms. Let's revisit our example:

• \((4r - 1)(7r + 2)\).

Expanding this polynomial involves multiplying each term in the first parenthesis with every term in the second:

• \(4r \cdot 7r + 4r \cdot 2 - 1 \cdot 7r - 1 \cdot 2\).

The result is a polynomial

• \(28r^2 + 8r - 7r - 2\),

which further simplifies to

• \(28r^2 + r - 2\).

Polynomial expansion is used extensively in algebra to manipulate expressions into a more usable form, helping to solve equations and understand polynomial functions better.