How do you factor difference of cubes? This is a common question among students studying algebra, as factoring polynomials can be a challenging task. The difference of cubes is a specific type of polynomial that can be factored using a specific formula. In this article, we will explore the steps involved in factoring the difference of cubes and provide some examples to illustrate the process.
The difference of cubes is a polynomial that takes the form \(a^3 – b^3\), where \(a\) and \(b\) are real numbers. The goal of factoring this type of polynomial is to express it as a product of two binomials. To achieve this, we can use the following formula:
\[a^3 – b^3 = (a – b)(a^2 + ab + b^2)\]
This formula is derived from the algebraic identity known as the sum of cubes. By applying this identity to the difference of cubes, we can factor the polynomial into two binomials.
Let’s take a look at an example to better understand the process:
Example: Factor the polynomial \(8x^3 – 27\).
First, we identify the values of \(a\) and \(b\) in the polynomial. In this case, \(a = 2x\) and \(b = 3\). Now, we can apply the formula for factoring the difference of cubes:
\[8x^3 – 27 = (2x – 3)(4x^2 + 6x + 9)\]
In this example, we have successfully factored the difference of cubes into two binomials.
It’s important to note that not all polynomials can be factored using the difference of cubes formula. The polynomial must have the form \(a^3 – b^3\) to be applicable. If the polynomial does not meet this condition, other factoring methods may need to be considered.
In conclusion, factoring the difference of cubes involves using the formula \(a^3 – b^3 = (a – b)(a^2 + ab + b^2)\). By identifying the values of \(a\) and \(b\) in the polynomial, we can apply this formula to factor the polynomial into two binomials. Practice and familiarity with the formula will help students become more proficient in factoring the difference of cubes.
