Problem 83 - Algebra and Trigonometry with Analytic Geometry

Question

Factor the polynomial. $$ 125-27 x^{3} $$

Step-by-Step Solution

Answer

The polynomial is factored as \((5 - 3x)(9x^2 + 15x + 25)\).

Step 1: Identify the formula for difference of cubes

The given polynomial can be recognized as the difference of two cubes: \[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \]. To use this formula, identify \( a \) and \( b \) such that \( a^3 = 125 \) and \( b^3 = 27x^3 \).

Step 2: Determine values of \( a \) and \( b \)

We have \( a^3 = 125 \), which implies \( a = 5 \), since \( 5^3 = 125 \). Similarly, for \( b^3 = 27x^3 \), \( b = 3x \) since \( (3x)^3 = 27x^3 \).

Step 3: Apply the difference of cubes formula

Now substitute \( a = 5 \) and \( b = 3x \) into the formula: \[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \] This gives us: \[ (5) - (3x)((5)^2 + 5(3x) + (3x)^2) \].

Step 4: Simplify into factors

Calculate each component: \(5^2 = 25\), \(5 \times 3x = 15x\), \((3x)^2 = 9x^2\). Now substitute these back:\[ (5 - 3x)(25 + 15x + 9x^2) \].

Step 5: Write down the fully factored form

Based on the previous calculations, the fully factored form of the polynomial is:\[ (5 - 3x)(9x^2 + 15x + 25) \].

Key Concepts

Factoring Polynomials

Factoring polynomials is a key skill in algebra that helps simplify expressions and solve equations. It involves expressing a polynomial as a product of its factors. To factor a polynomial, you first identify the type of polynomial you are working with, such as differences of cubes, quadratics, or others.

With the polynomial given in our exercise, we have a difference of cubes. This means we can rewrite the expression from its original form into two binomials and a trinomial.

Steps involved when factoring a polynomial:

• Identify the structure of the polynomial (e.g., difference of squares, cubes).
• Recognize and apply the appropriate factoring formula.
• Simplify each part of the factored expression.

Understanding how to factor polynomials helps in both simplifying expressions and solving polynomial equations in algebra.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations (like addition and multiplication) that define a specific quantity. They are everywhere in algebra and form the basics of algebraic operations.

In our problem, we dealt with the expression \(125 - 27x^3\), which was transformed using identifying elements within the expression. Here, 125 is recognized as the cube of 5, and \(27x^3\) as the cube of \(3x\). This identification is crucial for applying the difference of cubes formula.

To work with algebraic expressions effectively:

• Understand how each term is formed, including coefficients and powers.
• Identify common patterns such as cubes, squares, or other polynomial identities.
• Use strategic algebraic manipulations to simplify or rearrange terms.

Recognizing these elements is key to working through and simplifying complex expressions.

Polynomial Identities

Polynomial identities are equations that hold true for any values substituted into them. They form the backbone of factoring and simplifying polynomials, allowing us to see structural patterns and leverage them for simplification or solving.

In the exercise provided, the key polynomial identity utilized was the difference of cubes identity:\[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \]This identity recognizes that any expression of the form \(a^3 - b^3\) can be factored into a simpler product of a binomial and a trinomial.

Steps to use polynomial identities effectively:

• Identify the type of identity (difference of cubes, sum of cubes, etc.).
• Recognize the terms that match these identities in your expression.
• Apply the identity by substituting the defined values into the formula.

Understanding polynomial identities not only helps in factoring, but also provides deeper insights into the inherent structure and relationships within polynomials.