Question

Factor each polynomial completely. If the polynomial cannot be factored, say it is prime. $$ (5 x+1)^{3}-1 $$

Short answer

(5x + 1)^3 - 1 = 5x (25x^2 + 15x + 3).

Step-by-step solution

Recognize the formula

Identify the polynomial as a difference of cubes: \( (5x + 1)^3 - 1 = (a^3 - b^3) \) where \( a = 5x + 1 \) and \( b = 1 \).

Apply the difference of cubes formula

Use the difference of cubes factorization: \( a^3 - b^3 = (a - b)(a^2 + ab + b^2) \). Substitute \( a = 5x + 1 \) and \( b = 1 \) into the formula.

Substitute values

Substitute the values into the formula: \( (5x + 1)^3 - 1 = (5x + 1 - 1)((5x + 1)^2 + (5x + 1)(1) + 1^2) \).

Simplify the terms

Simplify each term: \( 5x + 1 - 1 = 5x \), \( (5x + 1)^2 = 25x^2 + 10x + 1 \), \( (5x + 1)(1) = 5x + 1 \), and \( 1^2 = 1 \).

Combine and simplify

Now combine and simplify the equation: \[ (5x)((25x^2 + 10x + 1) + (5x + 1) + 1) \] = \( (5x)((25x^2 + 10x + 1) + 5x + 1 + 1) \) = \[ (5x)(25x^2 + 15x + 3) \].

Key concepts

Understanding the Difference of Cubes

Knowing how to recognize and factor the difference of cubes is essential in algebra. The difference of cubes follows the formula: a^3 - b^3 = (a - b)(a^2 + ab + b^2) . Here, a and b are any terms, which, when cubed, give the original polynomial. In our exercise, we identify a = 5x + 1 and b = 1. This lets us transform our expression (5x + 1)^3 - 1 into a more workable form by substituting a and b into the formula. Practice by identifying and factoring different sets of a and b, ensuring strong comprehension of the formula's application.

Polynomial Factorization Techniques

Polynomial factorization is the process of breaking down a polynomial into simpler ‘factor’ polynomials that, when multiplied together, give back the original polynomial. Understanding various factorization techniques is key to tackling different polynomial forms. In the original exercise, we factored the expression using the difference of cubes formula. Here's a step-by-step recap:

• Identify a^3 - b^3 in the form (5x + 1)^3 - 1. Set a = 5x + 1 and b = 1.


• Apply the difference of cubes formula (a - b)(a^2 + ab + b^2).


• Simplify each term: a - b = 5x, (a^2 + ab + b^2) = (25x^2 + 10x + 1) + (5x + 1) + 1.


• Combine and simplify to get the final factored form: (5x)(25x^2 + 15x + 3)

This process can be replicated for various polynomials if the major concept is clear.

Simplifying Expressions in Algebra

Simplifying expressions allows us to rewrite them in a simpler or more useful form. This step is crucial for solving equations, factoring polynomials, and in other algebra problems. To simplify an expression, we need to perform operations such as combining like terms, factoring, using distributive properties, and reducing fractions. Let’s consider the simplification steps in the original exercise:

• First, identify terms to subtract, as in (a - b = 5x + 1 - 1 = 5x).


• Square terms like (5x + 1) which results in 25x^2 + 10x + 1.


• Multiply terms like (5x + 1)(1) to get 5x + 1, and cube 1^2 to get 1.
Combine all the simplified forms into one concise equation as (5x)(25x^2 + 15x + 3).

Practice breaking down expressions and simplifying each step to enhance problem-solving efficiency and accuracy.