Problem 33 - A Graphical Approach to College Algebra

Question

Find each product. $$2 b^{3}\left(b^{2}-4 b+3\right)$$

Step-by-Step Solution

Answer

The product is \(2b^{5} - 8b^{4} + 6b^{3}\).

Step 1: Distribute the First Term

To begin multiplying, distribute the first term of the first expression, \(2b^3\), across each term inside the parentheses. This involves multiplying \(2b^3\) by each term in the polynomial \((b^2 - 4b + 3)\).

Step 2: Multiply the First Pair of Terms

Multiply \(2b^3\) by \(b^2\): \[ 2b^3 imes b^2 = 2b^{5} \]

Step 3: Multiply the Second Pair of Terms

Multiply \(2b^3\) by \(-4b\): \[ 2b^3 imes (-4b) = -8b^{4} \]

Step 4: Multiply the Third Pair of Terms

Multiply \(2b^3\) by \(3\): \[ 2b^3 imes 3 = 6b^3 \]

Step 5: Combine All Products

Combine all the terms obtained from the multiplications: \[ 2b^{5} - 8b^{4} + 6b^{3} \]

Key Concepts

Distributive property

The distributive property is a fundamental concept in algebra that helps simplify expressions involving multiplication over addition or subtraction. It states that for any numbers or expressions, \( a(b + c) = ab + ac \).This means you can distribute the multiplier across terms within a parenthesis to break down complex expressions. In the context of the given exercise, the distributive property allows us to multiply the polynomial by distributing the term \( 2b^3 \) across each term inside the parentheses.

• The distributive property is used to eliminate parentheses by multiplying each term separately.
• This method ensures that every part of the expression is accounted for, leading to an accurate solution.

In practice, this property is especially useful for simplifying expressions and solving equations, making it one of the core building blocks for performing polynomial multiplication.

Algebraic expressions

Algebraic expressions are combinations of numbers, variables, and operations. They can represent anything from simple arithmetic to complex polynomials. Parts of an algebraic expression include terms, coefficients, and variables:

• Terms: Each part of an expression separated by a plus or minus sign, e.g., in the expression \( 2b^5 - 8b^4 + 6b^3 \) these are three distinct terms.
• Coefficients: The numerical part of a term, e.g., \( 2 \) in \( 2b^5 \).
• Variables: Symbols that represent numbers, e.g., \( b \).

Grasping algebraic expressions involves knowing how to manipulate and simplify them using arithmetic operations and properties like the distributive property. Understanding these expressions needs recognizing patterns and structures within them, enabling simplification, evaluation, or translation into equations to solve problems.

Exponents

Exponents in mathematics represent repeated multiplication of a number or variable by itself. In an expression such as \( b^3 \), the exponent \( 3 \) indicates that the base \( b \) is multiplied three times: \( b \, \cdot \, b \, \cdot \, b \).

• Basic Rules: Multiplication of the same bases involves adding the exponents, e.g., \( b^3 \, \times \, b^2 = b^{3+2} = b^5 \).
• Multiplication with coefficients: When a variable with an exponent is multiplied by a coefficient (like \( 2b^3 \)), each term's result keeps the coefficient intact while applying the exponent rules.

Exponents are crucial in indicating growth or scaling in algebraic expressions, making them integral to polynomial multiplication. Understanding how to handle exponents ensures that simplifications and calculations adhere to mathematical rules, ultimately leading to accurate outcomes.