Problem 5 - Algebra for College Students

Question

Find each indicated product. Remember the shortcut for multiplying binomials and the other special patterns we discussed in this section. $$8 a^{3} b^{4}\left(3 a b-2 a b^{2}+4 a^{2} b^{2}\right)$$

Step-by-Step Solution

Answer

The product is \(24a^4b^5 - 16a^4b^6 + 32a^5b^6\).

Step 1: Distribute Each Term

To multiply the polynomial, you need to distribute the monomial across each term within the parentheses. Consider the first term: \[8a^3b^4 \times 3ab = 24a^4b^5\].

Step 2: Multiply the Second Term

Next, distribute the monomial to the second term in the parentheses:\[8a^3b^4 \times -2ab^2 = -16a^4b^6\]. Make sure to include the negative sign.

Step 3: Multiply the Last Term

Finally, distribute the monomial to the last term:\[8a^3b^4 \times 4a^2b^2 = 32a^5b^6\].

Step 4: Combine the Results

Combine the results from each multiplication step to form the final expression:\[24a^4b^5 - 16a^4b^6 + 32a^5b^6\].

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that allows us to multiply a single term across each term in a polynomial within parentheses. This can be expressed mathematically as:

• If you have an expression like \(a(b + c + d)\), the distributive property helps you write it as \(ab + ac + ad\).

In the context of this exercise, you use the distributive property to multiply the monomial \(8a^3b^4\) with each term inside the parenthesis. This involves three separate multiplications: \(8a^3b^4 imes 3ab\), \(8a^3b^4 imes -2ab^2\), and \(8a^3b^4 imes 4a^2b^2\).
Remember to keep track of signs (positive and negative) while distributing, as seen in the step involving \(-2ab^2\). This property ensures your results are systematically derived and makes working with complex expressions much simpler.

Monomials

A monomial is a single term algebraic expression that consists of a constant, a variable, or a product of constants and variables. In this example, \(8a^3b^4\) is a monomial. Monomials are the simplest form of algebraic expressions.
Understanding how to manipulate monomials is crucial in algebra because they form the basic building blocks of more complex expressions like polynomials. When you

• Multiply two monomials, you multiply the coefficients (numerical parts) and then use the laws of exponents to multiply the variable parts.

For example, when you multiply \(8a^3b^4\) with \(3ab\), you multiply 8 by 3 to get 24 and \(a^3b^4\) by \(ab\) to get \(a^4b^5\) using the law \(a^m imes a^n = a^{m+n}\).
Mastering monomials is a step towards handling complex problems involving polynomials.

Polynomials

Polynomials are expressions that consist of multiple terms, which can include constants, variables, and exponents. Each term in a polynomial is either a monomial or a sum of monomials. In the given problem, \((3ab - 2ab^2 + 4a^2b^2)\) represents a polynomial that has three terms.
The degree of a polynomial is determined by the highest sum of the exponents of variables in any term.

• For example, in \(4a^2b^2\), the degree would be 4 (i.e., 2 + 2).

Understanding polynomials is essential because they form the basis of many algebraic expressions. They allow us to represent and solve equations that model real-world phenomena. By learning to multiply polynomials, as shown in this exercise, students gain powerful tools for predicting and finding unknowns.

Algebraic Expressions

Algebraic expressions consist of variables, numbers, and operations (such as addition, subtraction, multiplication, and division) put together in meaningful ways. They form the core language of algebra. Any combination of these elements that does not have an equality sign is considered an algebraic expression.
This exercise features several types of algebraic expressions:

• A single term, or monomial, like \(8a^3b^4\), and
• A polynomial, \(3ab - 2ab^2 + 4a^2b^2\).

Algebraic expressions can represent real quantities and relationships in equations, enabling students to solve problems by applying algebraic rules and properties, such as the distributive property used in multiplying these expressions.
Mastering how to work with algebraic expressions is fundamental for success in algebra and further mathematical studies, as they provide the structural framework for understanding equations and functions.