Problem 78 - Algebra and Trigonometry with Analytic Geometry

Question

Factor the polynomial. $$ 125 x^{3}-8 $$

Step-by-Step Solution

Answer

The polynomial \( 125x^3 - 8 \) factors to \( (5x - 2)(25x^2 + 10x + 4) \).

Step 1: Recognize the Form

The polynomial given is in the form of the difference of cubes: \[ 125x^3 - 8 = (5x)^3 - 2^3 \]. Identify it as a difference of cubes to use the appropriate factorization formula.

Step 2: Apply the Difference of Cubes Formula

The difference of cubes formula is \[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \]. Here, \( a = 5x \) and \( b = 2 \). Substitute these into the formula.

Step 3: Substitute into the Formula

Replace \( a \) and \( b \) with \( 5x \) and \( 2 \) respectively:\[ (5x)^3 - 2^3 = (5x - 2)((5x)^2 + (5x)(2) + 2^2) \].

Step 4: Simplify the Terms

Now simplify each term inside the second factor:- \((5x)^2 = 25x^2\)- \((5x)(2) = 10x\)- \(2^2 = 4\)Thus, the polynomial becomes \( (5x - 2)(25x^2 + 10x + 4) \).

Step 5: Verify the Factorization

Check by expanding: Multiply \((5x - 2)(25x^2 + 10x + 4)\) to verify whether it simplifies back to the original polynomial. It should confirm that the factorization is correct.

Key Concepts

polynomial factorization

Polynomial factorization is like breaking down a complex puzzle into smaller, easier pieces. It involves expressing a polynomial as a product of simpler polynomials. This process can be particularly helpful when solving equations or simplifying expressions. One key to polynomial factorization is recognizing special forms.

• Difference of Cubes: This is one such special form. It refers to expressions that can be written in the form of \( a^3 - b^3 \). Recognizing this form allows us to apply a specific formula to factor it effectively.

In the solution given for the polynomial \( 125x^3 - 8 \), we can identify it as a difference of cubes because it can be rewritten as \( (5x)^3 - (2)^3 \). With this understanding, we can apply the difference of cubes formula: \( a^3 - b^3 = (a - b)(a^2 + ab + b^2) \). This is crucial for students to master since it simplifies the factorization process substantially.

algebra

Algebra is a branch of mathematics that uses symbols and letters to represent numbers and quantities in equations and formulas. It forms the foundation for many advanced math topics and is crucial for developing problem-solving skills.

• Variables and Expressions: In algebra, expressions like \(125x^3 - 8\) become manageable through the use of variables, here represented by \(x\). Variables allow us to generalize solutions and solve problems for a wide range of numbers.
• Understanding Operations: Algebra also involves understanding the operations we perform on numbers and expressions, such as addition, subtraction, multiplication, and division. This includes recognizing patterns and structures, such as cubes and squares, which are central to the factorization taught in this exercise.

The application of algebra in tasks like polynomial factorization enables students to break down and solve more complex mathematical problems step-by-step. Each application serves as a fundamental building block for learning more advanced concepts in mathematics.

cubic equations

Cubic equations are polynomial equations where the highest power of the variable is three. In an equation like \(125x^3 - 8 = 0\), the goal is often to find the value of \(x\) that makes the equation true.

• Understanding Cubes: Recognizing cubic forms like \( a^3 \) is crucial. In the factorization process, knowing that \(125x^3\) can be expressed as \((5x)^3\) is vital for simplifying and solving the equation.
• Solving Cubic Equations: Once the polynomial is factored, such as into \((5x - 2)(25x^2 + 10x + 4) = 0\), solving the equation involves setting each factor equal to zero. For \(5x - 2 = 0\), solving for \(x\) gives one solution, while the quadratic \(25x^2 + 10x + 4 = 0\) may require further techniques, such as the quadratic formula, to find additional solutions.

Cubic equations often appear in diverse problem-solving scenarios. Understanding how to manage and manipulate them is a significant step in learning algebra.