Form 4

Basic Concept

The Cube of The Sum

\(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\)

 

The Cube of The Difference

\(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\)

Example 1

Factorize \(x^3 + 64\)

\begin{equation*}
\begin{split}
x^3 + 64& = x^3 + 4^3\\\\
x^3 + 64& = (x + 4)(x^2 + x \cdot 4 + 4^2)\\\\
x^3 + 64& = (x + 4)(x^2 + 4x + 16)
\end{split}
\end{equation*}

 

Example 2

Factorize \(125 - y^3\)

\begin{equation*}
\begin{split}
125 - y^3& = 5^3 - y^3\\\\
125 - y^3& = (5 - y)(5^2 + 5\cdot y + y^2)\\\\
125 - y^3& = (5 - y)(25 + 5y + y^2)
\end{split}
\end{equation*}

 

Example 3

Expand \((x + 5)(x^2 - 5x + 25)\)

\begin{equation*}
\begin{split}
(x + 5)(x^2 - 5x + 25)&= (x + 5)(x^2 - 5x + 5^2)\\\\
(x + 5)(x^2 - 5x + 25)& = x^3 + 5^3\\\\
(x + 5)(x^2 - 5x + 25)& = x^3 + 125
\end{split}
\end{equation*}

 

Example 4

Expand \((m - 3)(m^2 + 3m + 9)\)

\begin{equation*}
\begin{split}
(m - 3)(m^2 + 3m + 9) &= (m - 3)(m^2 + 3m + 3^2)\\\\
(m - 3)(m^2 + 3m + 9)& = m^3 - 3^3\\\\
(m - 3)(m^2 + 3m + 9)& = m^3 - 27
\end{split}
\end{equation*}

 

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