Problem 61 - Elementary and Intermediate Algebra

Question

Multiply. See Example 6. $$ \left(x^{2}+6 x+7\right)(2 x-5) $$

Step-by-Step Solution

Answer

The expanded expression is \(2x^3 + 7x^2 - 16x - 35\).

Step 1: Expand the Expression

To multiply the expressions \((x^2 + 6x + 7)(2x - 5)\), we need to distribute each term from the first binomial to each term in the second binomial. We can do this by using the distributive property.

Step 2: Multiply Each Term in the First Expression by Each Term in the Second

1. First, multiply \(x^2\) by the terms in \(2x - 5\): - \(x^2 \cdot 2x = 2x^3\) - \(x^2 \cdot -5 = -5x^2\)2. Next, multiply \(6x\) by the terms in \(2x - 5\): - \(6x \cdot 2x = 12x^2\) - \(6x \cdot -5 = -30x\)3. Finally, multiply \(7\) by the terms in \(2x - 5\): - \(7 \cdot 2x = 14x\) - \(7 \cdot -5 = -35\)

Step 3: Combine Like Terms

After distributing all terms, we have:\(2x^3 + (-5x^2) + 12x^2 + (-30x) + 14x + (-35)\).Now, combine like terms:- Combine the \(x^2\) terms: \(-5x^2 + 12x^2 = 7x^2\)- Combine the \(x\) terms: \(-30x + 14x = -16x\)This gives us the final expanded expression: \(2x^3 + 7x^2 - 16x - 35\).

Key Concepts

Distributive Property

The distributive property is a fundamental principle in algebra that helps in simplifying and solving expressions. It states that when you multiply a sum by a number, you can distribute the multiplication over each term inside the parentheses. For an expression like \((a + b)(c)\), using the distributive property means you multiply each term within the first parentheses by every term in the second. So, \(a \cdot c + b \cdot c\). This approach is essential when dealing with polynomial multiplication, as it breaks down complex expressions into simpler parts.

• Allows breaking down expressions easily
• Simplifies calculation by dealing with fewer numbers at a time

Applying the distributive property is crucial for success in algebra and essential for solving polynomial multiplication exercises.

Binomials

Binomials are polynomial expressions that consist of two terms, often separated by a plus or a minus sign. An example of a binomial is \((x + y)\) or \((2x - 5)\). Understanding how to handle binomials is key to mastering algebraic operations, such as addition, subtraction, and especially multiplication of polynomials.

• Two-term expressions make up binomials
• Typical structure involves a variable and constant

When multiplying two binomials, each term in one binomial must interact with each term in the other. This process involves leveraging the distributive property to ensure each term contributes to the resulting polynomial. Success with binomials opens doors to solving more complex polynomial expressions.

Expanding Expressions

Expanding expressions involves transforming more compact polynomial forms into an extended format where all terms are clearly shown. This is often necessary when multiplying two polynomials. For example, when expanding \((x^2 + 6x + 7)(2x - 5)\), each term in the first polynomial is multiplied by every term in the second to create a detailed expression.

• Reveals each computed term in a clear format
• Facilitates easy combination of like terms

This process requires precision and uses both distributive property and understanding of binomials. When you expand the expressions correctly, like terms, such as variables raised to the same power, can then be combined to simplify or solve the polynomial, yielding a complete and simplified expression. This method allows you to see and simplify the full polynomial expression with ease.