Problem 60 - Algebra 1: Concepts and Skills

Question

Find the product. $$ (3 a-2)(4 a+6) $$

Step-by-Step Solution

Answer

The product of the binomials \(3a - 2\) and \(4a + 6\) is \(12a^2 + 10a - 12\).

Step 1: Multiply the 'First' terms in the binomials.

The first terms of \(3a-2\) and \(4a+6\) are \(3a\) and \(4a\), respectively. These two terms are multiplied together resulting in: \(3a * 4a = 12a^2\). So, the first term of the product is \(12a^2\).

Step 2: Multiply the 'Outside' terms.

The outside terms are the first term of the first binomial and the second term of the second binomial, which are \(3a\) and \(6\), respectively. Multiplying these two together results in: \(3a * 6 = 18a\). So, the second term of the product is \(18a\).

Step 3: Multiply the 'Inside' terms.

The inside terms are the second term of the first binomial and the first term of the second binomial, which are \(-2\) and \(4a\), respectively. Multiplying these two together results in: \(-2 * 4a = -8a\). So, the third term of the product is \(-8a\).

Step 4: Multiply the 'Last' terms.

The last terms are the second terms of both binomials, which are \(-2\) and \(6\), respectively. Multiplying these two together results in: \(-2 * 6 = -12\). So, the fourth term of the product is \(-12\).

Step 5: Combine like terms.

Add all the products together and simplify any like terms: \(12a^2 + 18a - 8a - 12\) simplifies to \(12a^2 + 10a - 12\).

Key Concepts

The Essence of Binomials

Binomials are algebraic expressions that consist of exactly two terms. For example, in this exercise, the two binomials are \((3a - 2)\) and \((4a + 6)\). These are combined together through a multiplication process, which utilizes the distributive property to find the product.Understanding binomials is crucial because they frequently appear in algebra problems. They are often manipulated through operations like addition, subtraction, and multiplication. The standard form of a binomial might look like \(ax + b\), where \(a\) and \(b\) are constants and \(x\) is a variable.When working with binomials:

• Remember to handle each term inside them separately.
• Keep track of positive and negative signs, as they affect the final outcome of your calculations.

Multiplication Principles

In this exercise, multiplying binomials involves using the FOIL method, a handy mnemonic for remembering how to distribute the terms. FOIL stands for First, Outside, Inside, and Last:

• First: Multiply the first terms in each binomial.
• Outside: Multiply the outermost terms in the product.
• Inside: Multiply the inner terms.
• Last: Multiply the last terms in each binomial.

This method ensures that all combinations of terms are considered, resulting in an accurate product. For instance, multiplying \((3a - 2)\) and \((4a + 6)\), involves:

• First: \(3a \times 4a = 12a^2\)
• Outside: \(3a \times 6 = 18a\)
• Inside: \(-2 \times 4a = -8a\)
• Last: \(-2 \times 6 = -12\)

Notice how each step uses multiplication to combine terms, a key skill in solving polynomial problems.

Simplifying with Like Terms

Like terms are terms that share the same variable raised to the same power, making them combinable through addition or subtraction. In this exercise, after applying the FOIL method, you end up with terms of different types, including like terms.The terms are:

• \(12a^2\): This is a square term with \(a\) squared.
• \(18a\) and \(-8a\): These are like terms because they both have the variable \(a\).
• \(-12\): This is a constant term.

Combining like terms involves adding or subtracting their coefficients. For \(18a\) and \(-8a\), you get \(10a\) after combining. Hence, the final expression is simplified to:\[12a^2 + 10a - 12\] This simplification is crucial, as it not only tidies up your result but also makes further algebraic operations more manageable.