Problem 43 - Algebra 1: Concepts and Skills

Question

$$ (2 z+7)(3 z+2) $$

Step-by-Step Solution

Answer

The result is \( 6z^{2} + 25z + 14 \)

Step 1: Applying the FOIL Method: First terms multiplication

Multiply the first terms of each bracket. This gives: \( (2z \cdot 3z) = 6z^{2} \)

Step 2: Applying the FOIL Method: Multiplying Outside terms

Multiply the outside terms of both brackets together. This gives: \( (2z \cdot 2) = 4z \)

Step 3: Applying the FOIL Method: Multiplying Inside terms

Multiply the inside terms of both brackets together. This gives: \( (7 \cdot 3z) = 21z \)

Step 4: Applying the FOIL Method: Last terms multiplication

Multiply the last terms of each bracket. This gives \( (7 \cdot 2) = 14 \)

Step 5: Adding everything together

Combine all of the terms found in Steps 1-4. In this case: \( 6z^{2} + 4z + 21z + 14 \)

Step 6: Combining like terms

Combine the like terms, in this case, \( 4z \) and \( 21z \) to give: \( 6z^{2} + 25z + 14 \)

Key Concepts

Polynomial Multiplication

Polynomial multiplication involves multiplying two or more polynomials together to produce another polynomial. In algebra, this is a fundamental skill that helps in simplifying expressions and solving equations. The FOIL method is a common technique used to multiply two binomials. Here’s how it works:

• **First**: Multiply the first terms of each binomial.
• **Outside**: Multiply the outer terms in the binomials.
• **Inside**: Multiply the inner terms.
• **Last**: Multiply the last terms of each binomial.

By remembering the acronym FOIL, it becomes easier to follow the steps systematically and avoid missing any terms. This method ensures all parts of the binomial are multiplied together, which is crucial for accurate polynomial multiplication.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations such as addition, subtraction, multiplication, and division. Understanding these expressions is key to tackling problems across many mathematical topics, including polynomial multiplication.

When dealing with algebraic expressions, we often perform operations by treating variables as unknown values which hold place in mathematical operations. In the given exercise, the expression \((2z + 7)(3z + 2)\), for instance, involves letters or symbols that represent numbers, showing the abstract nature and flexibility of algebraic expressions.

Algebraic expressions enable us to perform versatile operations once we set specific values for the variables involved. They form the building blocks for equations and inequalities, which are fundamental in solving many real-world problems.

Like Terms

When simplifying algebraic expressions, the concept of like terms is essential. **Like terms** are terms in an algebraic expression that have identical variable parts, raised to the same power. You can only combine like terms to simplify expressions further.

In our solution, when we expanded the two binomials, we arrived at terms like \(4z\) and \(21z\). These both contain the variable \(z\) raised to the same power (in this case, power of 1), marking them as 'like terms'. This allows us to perform addition or subtraction across these terms. By combining them, \(4z + 21z = 25z\), the expression becomes simplified.

Recognizing and combining like terms is akin to gathering similar items together, making expressions much easier to manage and solve.

Binomials

Binomials are algebraic expressions that contain exactly two terms. Typically represented in the form \(a + b\) or \(a - b\), they are a special case of polynomials which can vastly simplify calculations when multiplied, added, or subtracted.

Using the FOIL method with binomials is particularly efficient, as it allows us to systematically break down and simplify expressions. In the example \((2z + 7)(3z + 2)\), each pair of terms across the two binomials is multiplied together systematically.

Recognizing binomials helps isolate specific areas in complex algebraic expressions to focus on, making larger problems more approachable and reducing errors in calculations. Binomials are foundational in algebra and offer plenty of practical applications in higher mathematics.