Problem 8 - Algebra 2 and Trigonometry

Question

Perform the indicated operations and write the result in simplest form. 3\(b(5 b-4)\)

Step-by-Step Solution

Answer

15b^2 - 12b

Step 1: Distribute the Outer Term

Start by distributing the constant 3 into the expression inside the parentheses. This means multiplying 3 with every term inside: 3\(b(5b - 4)\) becomes:3 \( imes \, b imes 5b - 3 imes b imes 4\)

Step 2: Calculate the Products

Now, perform the multiplication for each term:- First term: 3 \( imes \, b imes 5b = 15b^2\) - Second term: 3 \( imes \, b imes 4 = 12b\)

Step 3: Combine the Results

Combine the products from the previous step: The expression after distribution is: 15b^2 - 12b

Step 4: Simplify the Expression

Since there are no like terms to further combine, the expression 15b^2 - 12b is already in its simplest form.

Key Concepts

Polynomial Multiplication

Polynomial multiplication is a crucial concept in algebra, which involves multiplying polynomials to simplify or alter algebraic expressions.
To effectively multiply polynomials, one must apply the distributive property.When you have an expression such as 3\(b(5b-4)\), you need to multiply every term inside the parentheses by the term outside.

• Begin by distributing: Apply the term outside the parentheses (in this case, 3) to each term within the parentheses (5b and -4).
• Perform each individual multiplication: Multiply the term outside by each term inside separately, resulting in multiple smaller products.
• Repeat the process: If your expression has more terms inside the parentheses, keep repeating this process until you have distributed over all terms.

Remember that when multiplying polynomials, you multiply the coefficients (numerical parts) and then multiply the variable parts by adding the exponents if they are the same base.Exploring polynomial multiplication allows you to simplify expressions into a more digestible format, aiding further algebraic manipulation.

Simplifying Expressions

Simplifying expressions is the process of condensing an expression down to its simplest form, which often makes it easier to understand and work with.
In our example, after applying the distributive property, you end up with the expression: \(15b^2 - 12b\).
Here’s how the simplification process goes:

• Identify like terms: Like terms have the same variable raised to the same power. In our example, \(15b^2\) and \(-12b\) have different powers, so they are not like terms.
• Combine the like terms: If you have any, add or subtract them as necessary. Since \(15b^2\) and \(-12b\) are not like terms, they remain separate in the expression.
• Simplified as is: When no like terms can be combined, as with \(15b^2 - 12b\), you have your simplest form.

Simplifying expressions helps reduce errors in further calculations and allows you to see relationships between terms more clearly.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations (such as addition or multiplication). They form the basis for more complex mathematical observations and conclusions.
Understanding algebraic expressions involves recognizing patterns and relationships between the different components.

• Variables: Symbols that represent unknown or changing values, like \(b\) in the exercise.
• Coefficients: Numbers that multiply the variables, like 3 in \(3b(5b-4)\).
• Constants: Fixed numbers within an expression, which do not change, like -4 in the exercise.

Working with algebraic expressions requires familiarity with operations like polynomial multiplication and expression simplification.
As you work through algebraic problems, you'll often need to manipulate these expressions using various algebraic rules to find solutions to equations or to transform one expression into another.