Problem 1 - Elementary and Intermediate Algebra

Question

\(x^{3}+27\) is the ____ of two cubes and \(a^{3}-125\) is the difference of two ____.

Step-by-Step Solution

Answer

\(x^3 + 27\) is the sum of two cubes and \(a^3 - 125\) is the difference of two cubes.

Step 1: Understanding Sums and Differences of Cubes

The expression \( x^3 + 27 \) represents the sum of two cubes because it can be written as \( x^3 + 3^3 \). Similarly, the expression \( a^3 - 125 \) is the difference of two cubes since it can be written as \( a^3 - 5^3 \).

Step 2: Identifying Components

For \( x^3 + 27 \), the first cube is \( x^3 \) and the second cube is \( 3^3 \). For \( a^3 - 125 \), the first cube is \( a^3 \) and the second cube is \( 5^3 \).

Step 3: Recognizing Special Formulas

The formula for the sum of cubes is \( a^3 + b^3 = (a+b)(a^2 - ab + b^2) \), and the formula for the difference of cubes is \( a^3 - b^3 = (a-b)(a^2 + ab + b^2) \). These help identify whether an expression is a sum or difference of cubes.

Step 4: Completing the Blanks

Since \( x^3 + 27 \) is a sum of cubes, the blank is filled with 'sum'. For \( a^3 - 125 \), it is a difference of cubes, so the blank is filled with 'cubes'.

Key Concepts

Algebraic Expressions

Algebraic expressions are combinations of variables, coefficients, and constants. These components are linked together through operations such as addition, subtraction, multiplication, and division. Simply put, think of them as mathematical phrases you assemble from building blocks like numbers and symbols.
For example, in the expression \(x^3 + 27\), you have:

• Variables: These are symbols (like \(x\)) that can represent unknown values.
• Constants: These are fixed numbers, such as 27, which do not change.
• Coefficients: These are numbers placed before variables to denote multiplication; for instance, in \(3x\), the number 3 is a coefficient.
• Operations: These include addition (+), subtraction (-), multiplication (\(\times\)), and division (\(\div\)).

By understanding the roles of these parts, you can more easily manipulate and solve algebraic expressions. This knowledge is essential for recognizing patterns and applying formulas effectively.

Special Factorization Formulas

Special factorization formulas come in handy when you need to break down complex algebraic expressions into simpler factors. These formulas make certain types of expressions easier to work with by recognizing patterns such as the sum or difference of cubes.
Two critical special factorization formulas are:

• Sum of Cubes: \(a^3 + b^3 = (a+b)(a^2 - ab + b^2)\)
• Difference of Cubes: \(a^3 - b^3 = (a-b)(a^2 + ab + b^2)\)

These formulas are straightforward when you break them down:

• The first factor, \((a+b)\) or \((a-b)\), captures the simple addition or subtraction between the cube roots, \(a\) and \(b\).
• The second factor, \(a^2 - ab + b^2\) or \(a^2 + ab + b^2\), further simplifies the expression by eliminating the cubes but keeping the relationship between \(a\) and \(b\).

Recognizing when to apply these formulas can make solving algebraic problems much more manageable.

Cubic Polynomials

Cubic polynomials are expressions involving terms raised to the third power. They are critical in algebra because they often model real-world situations involving area, volume, and other three-dimensional relationships. The general form of a cubic polynomial is \(ax^3 + bx^2 + cx + d = 0\), where \(a\), \(b\), \(c\), and \(d\) are constants and \(a eq 0\).
Key features of cubic polynomials include:

• Degree: The degree is the highest power of the variable, which is 3 for cubic polynomials.
• Roots: These are the solutions to the polynomial equation when set to zero, representing values of \(x\) that satisfy the equation \(ax^3 + bx^2 + cx + d = 0\).
• Shape: On a graph, cubic polynomials have a distinctive curve that can have one real root or three real roots (although complex roots also come in conjugate pairs).

Understanding the properties of cubic polynomials is crucial for solving and graphing these expressions, as well as for using them to accurately model complex real-world scenarios.