Sum and Difference of Cubes are special cases of polynomial expressions that follow certain patterns, which can be factored using specific formulas. Understanding these patterns and formulas are helpful in solving equations.
A Sum of Cubes is a special product that factors a binomial being subtracted into the product of a binomial and a trinomial. It can be expressed and factored as such:
In order for a binomial to be considered a Sum of Cubes, the 2 terms must be perfect cubes (meaning their cube roots must be whole numbers). In addition, the first term, \(a\), must be postive and the second, \(b\), must be subtracted.
Fully factor \(x^3 + 8\).
First, we can identify \(a^3\) and \(b^3\):
\(a^3 = x^3\)
\(b^3 = 8\)
Next, we can identify \(a\) and \(b\) by calculating the cube roots of \(a^3\) and \(b^3\):
\(a = \sqrt[3]{x^3}\)
\(\textcolor{red}{a = x}\)
\(b = \sqrt[3]{8}\)
\(\textcolor{blue}{b = 8}\)
Then, we can substitute \(a\) and \(b\) into the Sum of Cubes formula:
\(\textcolor{red}{a}^3 + \textcolor{blue}{b}^3 = (\textcolor{red}{a}+\textcolor{blue}{b})(\textcolor{red}{a}^2 - \textcolor{red}{a}\textcolor{blue}{b} + \textcolor{blue}{b}^2) \)
\(\textcolor{red}{x}^3 + \textcolor{blue}{2}^3 = (\textcolor{red}{x}+\textcolor{blue}{2})(\textcolor{red}{x}^2 - (\textcolor{red}{x})(\textcolor{blue}{2}) + \textcolor{blue}{2}^2) \)
\(x^3 + 8 =(x+2)(x^2 - 2x + 4) \)
Therefore, we can determine that \( x^3 + 8 \) fully factored is \(\boldsymbol{(x+2)(x^2 - 2x + 4)}\).
A Difference of Cubes is a special product that factors a binomial being subtracted into the product of a binomial and a trinomial. It can be expressed and factored as such:
When you have the difference of two cubes \( a^3 - b^3 \), where \( a \) and \( b \) are real numbers the expression can be written as \( (a-b)(a^2 + ab + b^2) \).
Fully factor \( x^3 - 27 \).
First, we can identify \( a^3 \) and \( b^3 \):
\(a^3 = x^3\)
\(b^3 = 27 \)
Next, we can identify \(a\) and \(b\) by calculating the cube roots of \(a^3\) and \(b^3\):
\(a = \sqrt[3]{x^3}\)
\(\textcolor{red}{a = x}\)
\(b = \sqrt[3]{27}\)
\(\textcolor{blue}{b = 3}\)
Then, we can substitute \(a\) and \(b\) into the Difference of Cubes formula:
\(\textcolor{red}{a}^3 - \textcolor{blue}{b}^3 = (\textcolor{red}{a}-\textcolor{blue}{b})(\textcolor{red}{a}^2 + \textcolor{red}{a}\textcolor{blue}{b} + \textcolor{blue}{b}^2)\)
\(\textcolor{red}{x}^3 - \textcolor{blue}{3}^3 = (\textcolor{red}{x}-\textcolor{blue}{3})(\textcolor{red}{x}^2 + (\textcolor{red}{x})(\textcolor{blue}{3}) + \textcolor{blue}{3}^3)\)
\( x^3 - 27 = (x-3)(x^2 + 3x + 9) \)
Therefore, we can determine that \( x^3 - 27\) fully factored is \(\boldsymbol{(x+2)(x^2 - 2x + 4)}\).
\( x^3 + 125 \)
First, we can identify \( a^3 \) and \( b^3 \):
\(a^3 = x^3\)
\(b^3 = 125\)
Next, we can identify \(a\) and \(b\) by calculating the cube roots of \(a^3\) and \(b^3\):
\(a = \sqrt[3]{x^3}\)
\(\textcolor{red}{a = x}\)
\(b = \sqrt[3]{125}\)
\(\textcolor{blue}{b = 5}\)
Then, we can substitute \(a\) and \(b\) into the Sum of Cubes formula:
\(\textcolor{red}{a}^3 + \textcolor{blue}{b}^3 = (\textcolor{red}{a}+\textcolor{blue}{b})(\textcolor{red}{a}^2 - \textcolor{red}{a}\textcolor{blue}{b} + \textcolor{blue}{b}^2) \)
\(\textcolor{red}{x}^3 + \textcolor{blue}{5}^3 = (\textcolor{red}{x}+\textcolor{blue}{5})(\textcolor{red}{x}^2 - (\textcolor{red}{x})(\textcolor{blue}{5}) + \textcolor{blue}{5}^2) \)
\(x^3 + 125 =(x+5)(x^2 - 5x + 25) \)
Therefore, we can determine that \(x^3 + 125\) fully factored is \(\boldsymbol{x^3 + 125 =(x+5)(x^2 - 5x + 25)}\).
\(x^3 - 343\)
First, we can identify \(a^3\) and \(b^3\):
\(a^3 = x^3\)
\(b^3 = 343\)
Next, we can identify \(a\) and \(b\) by calculating the cube roots of \(a^3\) and \(b^3\):
\(a = \sqrt[3]{x^3}\)
\(\textcolor{red}{a = x}\)
\(b = \sqrt[3]{343}\)
\(\textcolor{blue}{b = 7}\)
Finally, we can substitute \(a\) and \(b\) into the Difference of Cubes formula:
\(\textcolor{red}{a}^3 - \textcolor{blue}{b}^3 = (\textcolor{red}{a}-\textcolor{blue}{b})(\textcolor{red}{a}^2 + \textcolor{red}{a}\textcolor{blue}{b} + \textcolor{blue}{b}^2)\)
\(\textcolor{red}{x}^3 - \textcolor{blue}{7}^3 = (\textcolor{red}{x}-\textcolor{blue}{7})(\textcolor{red}{x}^2 + (\textcolor{red}{x})(\textcolor{blue}{7}) + \textcolor{blue}{7}^2)\)
\( x^3 - 343 =(x-7)(x^2 + 7x + 49) \)
Therefore, we can determine that \( x^3 - 343 \) fully factored is \(\boldsymbol{x^3 - 343 =(x-7)(x^2 + 7x + 49)}\).