Question

Sum or Difference of Two Cubes. $$343-27 y^{3}$$

Short answer

(7 - 3y)(49 + 21y + 9y^2)

Step-by-step solution

Identify the Cubes

Identify the cube terms in the expression. Here, 343 and 27 are perfect cubes as 343 can be written as 7^3 and 27 can be written as 3^3.

Convert to Variables

Express the given expression in terms of cube of variables. So, 343 can be expressed as (7)^3 and 27y^3 as (3y)^3.

Apply the Difference of Cubes Formula

Apply the formula a^3 - b^3 = (a - b)(a^2 + ab + b^2), where a = 7 and b = 3y to factor the expression.

Factor the Expression

By substituting a and b into the formula, the expression becomes (7 - 3y)((7)^2 + (7)(3y) + (3y)^2).

Simplify the Factors

Simplify the factors to get (7 - 3y)(49 + 21y + 9y^2).

Key concepts

Factoring Polynomials

Factoring polynomials is a crucial skill in algebra that allows students to break down complex expressions into simpler, more manageable parts. It involves rewriting a polynomial as a product of its factors, akin to splitting a number into its multiplicative pieces. For instance, when you are given a polynomial like a cubic function—represented by an expression like ax^3 + bx^2 + cx + d—factoring could involve identifying and extracting common factors or using special formulas to simplify the expression.

When dealing with a difference of two cubes, such as 343 - 27y^3, you're essentially aiming to identify the cubic roots of each term to rewrite the expression based on a standard algebraic identity. Understanding the basic form and coefficients of polynomials is key to recognizing patterns that suggest certain factoring techniques, like the one used for the difference of two cubes. As you practice factoring polynomials, you'll become more adept at seeing these patterns quickly and applying the correct methods to simplify the polynomial expressions you encounter.

Cubic Functions

Cubic functions are polynomial functions where the highest degree of the variable is three, commonly written in the general form f(x) = ax^3 + bx^2 + cx + d, where a, b, c, and d are constants and a is nonzero. These functions are recognizable by their characteristic curved shape, which can feature one or two bends, resulting in varying behavior such as having one, two, or three real roots.

The a term in the cubic function determines the width and direction of the graph, as well as indicating whether the function's ends will travel off in the same or opposite directions. To better understand cubic functions and how to manipulate them algebraically, it is helpful to explore specific cases, like the sum or difference of two cubes. By mastering this, one gains deeper insight into the nature of cubic equations and their graphical representations, enabling the solution of real-world problems modeled by cubic relationships.

Algebraic Identities

Algebraic identities are equations that are universally true for all values of the variables involved. They serve as tools that simplify expressions and solve equations more efficiently. A common example of an algebraic identity is the difference of two squares, a^2 - b^2 = (a + b)(a - b), which provides a simple method to factor expressions with squared terms.

Another powerful identity used for cubic functions is the difference of two cubes, represented by a^3 - b^3 = (a - b)(a^2 + ab + b^2). This identity is particularly useful in factoring expressions like 343 - 27y^3, since it provides a systematic approach to breaking down cubic terms into a product of a binomial and a trinomial. Knowing these identities and how to apply them can greatly streamline the process of manipulating algebraic expressions, enabling students to solve complex problems with greater ease and confidence.