Problem 30 - Elementary and Intermediate Algebra

Question

Multiply. See Example 2. $$ -4 b^{3}\left(2 b^{2}-2 b+2\right) $$

Step-by-Step Solution

Answer

The expression simplifies to \(-8b^5 + 8b^4 - 8b^3\).

Step 1: Distribute the Term

Begin by distributing the term \(-4b^3\) through each term inside the parenthesis \((2b^2 - 2b + 2)\). This means we multiply \(-4b^3\) by each of the terms inside: \(2b^2\), \(-2b\), and \(2\).

Step 2: Multiply \(-4b^3\) with \(2b^2\)

Multiply the coefficients \(-4\) and \(2\), which gives \(-8\). Then multiply the powers of \(b\): \(b^3\) and \(b^2\) which gives \(b^{5}\). Thus the result is \(-8b^5\).

Step 3: Multiply \(-4b^3\) with \(-2b\)

Multiply the coefficients \(-4\) and \(-2\), which gives \(8\). Then multiply the powers of \(b\): \(b^3\) and \(b\) to get \(b^{4}\). Thus the result is \(8b^4\).

Step 4: Multiply \(-4b^3\) with \(2\)

Here, only the coefficients are multiplied \(-4\) and \(2\) which gives \(-8\). There is no \(b\) to multiply with, so the result remains \(-8b^3\).

Step 5: Combine all parts together

Now, combine all the resulting terms from each multiplication step to form the expanded expression: \(-8b^5 + 8b^4 - 8b^3\).

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that allows us to simplify expressions and solve equations. It states that for any numbers or expressions, multiplying a single term by a sum or difference inside parentheses is the same as multiplying each term inside the parentheses by the single term outside. This property helps us break down complex expressions into simpler ones.For example, if we have the expression \(-4b^3(2b^2 - 2b + 2)\), we can use the distributive property to multiply \(-4b^3\) by each term inside the parentheses separately:

• Multiply \(-4b^3\) by \(2b^2\)
• Multiply \(-4b^3\) by \(-2b\)
• Multiply \(-4b^3\) by \(2\)

By applying the distributive property in this way, we expand the expression to find its equivalent form. It makes working with polynomials much more manageable and is a critical tool for any algebra student.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and mathematical operations. They are the building blocks of algebra and help represent real-world situations and mathematical problems in a symbolic form. Variables, often represented by letters such as \(b\), stand for unknown or generalized numbers and can be manipulated through operations.An algebraic expression like \(-4b^3(2b^2 - 2b + 2)\) consists of terms that are multiplied or added together. This particular expression involves both multiplication and addition within the parentheses. The distributive property allows us to handle such expressions effectively by breaking down the multiplication into simpler steps.Understanding how to work with algebraic expressions enables us to solve equations, simplify expressions, and calculate values that these expressions represent. It's essential to practice recognizing and rearranging these components to become proficient in algebra.

Monomials

Monomials are algebraic expressions that consist of a single term. They include a coefficient (a number) and one or more variables raised to a power, like \(-4b^3\). Each component of a monomial serves a specific purpose:

• Coefficient: the numerical part that is multiplied by the variable(s).
• Variables: letters representing different numbers or values.
• Exponents: numbers indicating how many times the variable is multiplied by itself.

In our example, \(-4b^3\) is a monomial where \(-4\) is the coefficient, \(b\) is the variable, and \(3\) is the exponent. Recognizing monomials is critical because they often appear in larger algebraic expressions, and knowing how to manipulate them allows us to perform operations like distribution effectively.

Exponents

Exponents are powerful mathematical tools that simplify the representation of repeated multiplication. In an expression such as \(b^3\), the number \(3\) is the exponent, and it tells us that the base, \(b\), is multiplied by itself three times: \(b \times b \times b\).Exponents have specific rules that make calculations easier:

• Multiplying same bases: When multiplying terms with the same base, you add the exponents: \(b^m \times b^n = b^{m+n}\).
• Power of a power: When raising a power to another power, multiply the exponents: \((b^m)^n = b^{m \cdot n}\).
• Zero exponent: Any non-zero number raised to the exponent of zero is 1: \(b^0 = 1\).

In the problem given, we applied the exponent rule that combines powers when multiplying, allowing us to quickly determine the result of expressions like \(-4b^3 \times 2b^2\) as \(-8b^5\). Mastery of exponents is essential for simplifying expressions and solving algebraic problems efficiently.