Chapter 10: Problem 82
Find the product. $$(3 a-2)(4 a+6)$$
Short Answer
Expert verified
The product of the binomials \((3a - 2)(4a + 6)\) is \(12a^2 + 10a - 12\).
Step by step solution
01
FOIL Step - First terms
Multiply the first terms in each binomial: \(3a * 4a = 12a^2\)
02
FOIL Step - Outer terms
Multiply the outer terms of the binomials: \(3a * 6 = 18a\)
03
FOIL Step - Inner terms
Multiply the inner terms of the binomials: \(-2 * 4a = -8a\)
04
FOIL Step - Last terms
Multiply the last terms of the binomials: \(-2 * 6 = -12\)
05
Combine like terms
Combine the like terms (in this case, the terms with variable \(a\)). The expression becomes: \(12a^2 + 18a - 8a -12\) which simplifies to: \(12a^2 + 10a - 12\)
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Multiplying Binomials
Understanding how to multiply binomials is a fundamental skill in algebra. Binomials are algebraic expressions containing two terms, such as \(3a - 2\) and \(4a + 6\), separated by either a plus or minus sign.
When multiplying two binomials, the FOIL method comes in handy. FOIL stands for First, Outer, Inner, Last, referring to the position of the terms within each binomial that you need to multiply together. Each multiplication produces a term, and these terms are then added together to get the final product. Keeping these steps organized will help ensure that all combinations are accounted for and that the multiplication is thorough. Let's break this down with an example:
When multiplying two binomials, the FOIL method comes in handy. FOIL stands for First, Outer, Inner, Last, referring to the position of the terms within each binomial that you need to multiply together. Each multiplication produces a term, and these terms are then added together to get the final product. Keeping these steps organized will help ensure that all combinations are accounted for and that the multiplication is thorough. Let's break this down with an example:
- First terms: Multiply the first term of each binomial.
- Outer terms: Multiply the outer terms of the binomials.
- Inner terms: Multiply the inner terms of the binomials.
- Last terms: Multiply the last terms of each binomial.
Combining Like Terms
Once you have used the FOIL method to expand the product of two binomials, you will often be left with an expression that needs further simplification. This is where combining like terms comes into play.
Like terms are terms that have the same variable raised to the same power. In the step-by-step example, after applying the FOIL method, we get the terms \(12a^2\), \(18a\), \(−8a\), and \(−12\). The terms \(18a\) and \(−8a\) are like terms because they both contain the variable \(a\) raised to the power of one.
To combine them, you add or subtract their coefficients (the numbers in front of the variables): \(18a - 8a = 10a\). Combining like terms is an essential step in simplifying algebraic expressions and ensuring that the final answer is as clear and concise as possible.
Like terms are terms that have the same variable raised to the same power. In the step-by-step example, after applying the FOIL method, we get the terms \(12a^2\), \(18a\), \(−8a\), and \(−12\). The terms \(18a\) and \(−8a\) are like terms because they both contain the variable \(a\) raised to the power of one.
To combine them, you add or subtract their coefficients (the numbers in front of the variables): \(18a - 8a = 10a\). Combining like terms is an essential step in simplifying algebraic expressions and ensuring that the final answer is as clear and concise as possible.
Polynomial Multiplication
When multiplying binomials, we are essentially performing a type of polynomial multiplication. A polynomial is an algebraic expression that can have one or more terms. The example we are working with, \(3a - 2)(4a + 6)\), involves multiplying two binomials, each of which is a simple polynomial.
Polynomial multiplication involves distributing each term of one polynomial to every term of the other polynomial. The FOIL method is one strategy that applies specifically to binomials. However, for polynomials with more than two terms, you would use the distributive property repeatedly until every term has been combined. This is also referred to as the 'area' method or 'box' method, where you create a grid to systematically compute the products.
Understanding how to perform polynomial multiplication accurately is crucial for solving more complex algebraic equations and simplifying expressions with higher-degree polynomials.
Polynomial multiplication involves distributing each term of one polynomial to every term of the other polynomial. The FOIL method is one strategy that applies specifically to binomials. However, for polynomials with more than two terms, you would use the distributive property repeatedly until every term has been combined. This is also referred to as the 'area' method or 'box' method, where you create a grid to systematically compute the products.
Understanding how to perform polynomial multiplication accurately is crucial for solving more complex algebraic equations and simplifying expressions with higher-degree polynomials.
Algebraic Expressions
Algebraic expressions are combinations of variables, numbers, and operations such as addition and subtraction. In the context of our problem, the expression \(3a - 2)(4a + 6)\) is an example of a product of two algebraic expressions or binomials. It combines constants, coefficients, and variables.
Mastering the manipulation of algebraic expressions, including how to expand, simplify, and factor them, is key to advancing in mathematics. This involves comprehending the order of operations, the distributive property, and other algebraic principles. Each expression presents its own set of challenges, but with practice and understanding of the underlying concepts, working with algebraic expressions becomes a fundamental skill in a student's mathematical toolkit.
Mastering the manipulation of algebraic expressions, including how to expand, simplify, and factor them, is key to advancing in mathematics. This involves comprehending the order of operations, the distributive property, and other algebraic principles. Each expression presents its own set of challenges, but with practice and understanding of the underlying concepts, working with algebraic expressions becomes a fundamental skill in a student's mathematical toolkit.
