7.11: Factorization of Special Cubics
Factor the following cubic polynomial: \begin{align*}375x^3+648\end{align*}.
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James Sousa: Factoring Sum and Difference of Cubes
Guidance
While many cubics cannot easily be factored, there are two special cases that can be factored quickly. These special cases are the sum of perfect cubes and the difference of perfect cubes.
- Factoring the sum of two cubes follows this pattern: \begin{align*}x^3+y^3=(x+y)(x^2-xy+y^2)\end{align*}
- Factoring the difference of two cubes follows this pattern: \begin{align*}x^3-y^3=(x-y)(x^2+xy+y^2)\end{align*}
The acronym SOAP can be used to help you remember the positive and negative signs when factoring the sum and difference of cubes. SOAP stands for "Same", "Opposite", "Always Positive". "Same" refers to the first sign in the factored form of the cubic being the same as the sign in the original cubic. "Opposite" refers to the second sign in the factored cubic being the opposite of the sign in the original cubic. "Always Positive" refers to the last sign in the factored form of the cubic being always positive. See below:
Example A
Factor: \begin{align*}x^3+27\end{align*}.
Solution: This is the sum of two cubes and uses the factoring pattern: \begin{align*}x^3+y^3=(x+y)(x^2-xy+y^2)\end{align*}.
\begin{align*}x^3+3^3=(x+3)(x^2-3x+9)\end{align*}.
Example B
Factor: \begin{align*}x^3-343\end{align*}.
Solution: This is the difference of two cubes and uses the factoring pattern: \begin{align*}x^3-y^3=(x-y)(x^2+xy+y^2)\end{align*}.
\begin{align*}x^3-7^3=(x-7)(x^2+7x+49)\end{align*}.
Example C
Factor: \begin{align*}64x^3-1\end{align*}.
Solution: This is the difference of two cubes and uses the factoring pattern: \begin{align*}x^3-y^3=(x-y)(x^2+xy+y^2)\end{align*}.
\begin{align*}(4x)^3-1^3=(4x-1)(16x^2+4x+1)\end{align*}.
Concept Problem Revisited
Factor the following cubic polynomial: \begin{align*}375x^3+648\end{align*}.
First you need to recognize that there is a common factor of \begin{align*}3\end{align*}. \begin{align*}375x^3+648=3(125x^3+216)\end{align*}
Notice that the result is the sum of two cubes. Therefore, the factoring pattern is \begin{align*}x^3+y^3=(x+y)(x^2-xy+y^2)\end{align*}.
\begin{align*}375x^3 +648 = 3(5x+6)(25x^2-30x+36)\end{align*}
Vocabulary
- Difference of Two Cubes
- The difference of two cubes is a special polynomial in the form of \begin{align*}x^3-y^3\end{align*}. This type of polynomial can be quickly factored using the pattern: \begin{align*}(x^3-y^3)=(x-y)(x^2+xy+y^2)\end{align*}
- Sum of Two Cubes
- The sum of two cubes is a special polynomial in the form of \begin{align*}x^3+y^3\end{align*}. This type of polynomial can be quickly factored using the pattern: \begin{align*}(x^3+y^3)=(x+y)(x^2-xy+y^2)\end{align*}
Guided Practice
Factor each of the following cubics.
1. \begin{align*}x^3+512\end{align*}
2. \begin{align*}8x^3+125\end{align*}
3. \begin{align*}x^3-216\end{align*}
Answers:
1. \begin{align*}x^3+8^3=(x+8)(x^2-8x+64)\end{align*}.
2. \begin{align*}(2x)^3+5^3=(2x+5)(4x^2-10x+25)\end{align*}.
3. \begin{align*}x^3-6^3=(x-6)(x^2+6x+36)\end{align*}.
Practice
Factor each of the following cubics.
- \begin{align*}x^3+h^3\end{align*}
- \begin{align*}a^3+125\end{align*}
- \begin{align*}8x^3+64\end{align*}
- \begin{align*}x^3+1728\end{align*}
- \begin{align*}2x^3+6750\end{align*}
- \begin{align*}h^3-64\end{align*}
- \begin{align*}s^3-216\end{align*}
- \begin{align*}p^3-512\end{align*}
- \begin{align*}4e^3-32\end{align*}
- \begin{align*}2w^3-250\end{align*}
- \begin{align*}x^3+8\end{align*}
- \begin{align*}y^3-1\end{align*}
- \begin{align*}125e^3-8\end{align*}
- \begin{align*}64a^3+2197\end{align*}
- \begin{align*}54z^3+3456\end{align*}
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Here you'll learn to factor the sum and difference of perfect cubes.