Problem 6 - Precalculus : Building Concepts and Connections

Question

Complete them to review topics relevant to the remaining exercises. Multiply: \(x^{3}\left(x^{2}-3\right)(x+1)\).

Step-by-Step Solution

Answer

The simplified result of the multiplication is \(x^{6} + x^{5} -3x^{4} -3x^{3}\).

Step 1: Distribute \(x^{3}\) to \((x^{2}-3)\)

Multiply each term of \((x^{2}-3)\) by \(x^{3}\) to get: \(x^{5}-3x^{3}\).

Step 2: Distribute \(x^{5}-3x^{3}\) to \((x+1)\)

Then, multiply each term of \((x+1)\) by each term of \(x^{5}-3x^{3}\) to get: \(x^{6} + x^{5} -3x^{4} -3x^{3}\).

Step 3: Simplify the result

The answer should be in its simplest form. In this case, there are no like terms to combine, so the answer is \(x^{6} + x^{5} -3x^{4} -3x^{3}\).

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra, a method used to simplify expressions by distributing a multiplied value across terms inside a set of parentheses. When you see an expression like \( a(b + c) \), you apply the distributive property by multiplying \( a \) with each term inside the parenthesis separately, resulting in \( ab + ac \).
To break it down:

• Multiply \( a \) with \( b \)
• Multiply \( a \) with \( c \)

The distributive property lets you expand expressions like \( x^3(x^2 - 3) \). Multiply \( x^3 \) by \( x^2 \) to get \( x^5 \), and \( x^3 \) by \(-3 \) to get \(-3x^3 \). This transformation helps in further multiplication and simplification of expressions.

Algebraic Expressions

Algebraic expressions consist of numbers, variables, and operations like addition and multiplication. They serve as a key element in algebra, allowing us to construct equations and showcase relationships between numbers and variables. Consider the expression \( x^3(x^2 - 3)(x + 1) \), it consists of terms:

• \( x^3 \) - a single term with a variable, raised to a power
• \( x^2 - 3 \) - a difference of a squared variable and a constant
• \( x + 1 \) - a sum of a variable and a constant

Each term in an algebraic expression can be modified using rules and operations to simplify or solve equations. Mastery of manipulating these expressions is crucial for solving polynomial multiplication problems.

Exponent Rules

Exponent rules are useful mathematical shortcuts that facilitate operations with expressions involving powers. They simplify the multiplication, division, and transformation of powers. When dealing with polynomial multiplication like in the expression \( x^3(x^2 - 3)(x + 1) \), exponent rules come in handy. Here's how they work:

• Multiplying Powers with the Same Base: Add the exponents together. For example, \( x^3 \times x^2 = x^{3+2} = x^5 \).
• Power of a Power: Multiply the exponents. For instance, \( (x^2)^3 = x^{2\times3} = x^6 \).
• Power of a Product: Distribute the exponent to each factor in the product. Example: \( (ab)^n = a^n \cdot b^n \).

By applying these rules correctly, complex polynomial expressions can be greatly simplified, enabling easier computation and solution finding.