What formula do I use for factoring these?

Question

An elementary question, but I am having a lot of discrepancies identifying the correct formula to use, I can do more complex ones but not the simple ones if that makes sense.

a) $8x^3 + 1$

b) $m^2 - 100n^2$

Thank you, regards.

Answer 1

In general, $$a^3+b^3 = (a+b)(a^2-ab+b^2)$$ $$a^3-b^3 = (a-b)(a^2+ab+b^2)$$ $$a^2-b^2 =(a+b)(a-b)$$

Answer 2

Hints:

a) $A^3+B^3=(A+B)(A^2-AB+B^2)$

b) $A^2-B^2=(A-B)(A+B)$.

(Verify these equalities first by performing the multiplications on the right.)

Answer 3

You have to know the sum of cubes formula, and the differences of squares formula. $$a^3+b^3=(a+b)(a^2-ab+b^2)$$ $$a^2-b^2=(a+b)(a-b)$$ We can apply these to our expressions. $$8x^3+1=(2x)^3+1^3$$ $$=(2x+1)(4x^2-2x+1)$$ Now for the second one: $$m^2-100n^2=m^2-(10n)^2$$ $$=(m+10n)(m-10n)$$ Hope I helped!

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Here is a way to help you remember the formulas (more specifically, what signs to use in the factorization): $$a^3\color{green}+b^3=(a\color{green}+b)(a^2\color{red}-ab\color{green}+b^2)$$ $$a^3\color{red}-b^3=(a\color{red}-b)(a^2\color{green}+ab\color{green}+b^2)$$ $$a^2\color{red}-b^2=(a\color{green}+b)(a\color{red}-b)$$ Now you may be wondering, "How do you factor $a^2+b^2$?" Well, you cannot factor using real numbers only. You have to use complex numbers. The formula is: $$a^2\color{green}+b^2=(a\color{green}+bi)(a\color{red}-bi)$$ where $i$ is equal to $\sqrt{-1}$. Think of it as a difference of squares, but with an $i$ after the second number.

Answer 4

$\!\begin{eqnarray} {\bf Hint}\ \ \ \color{#c00}m - \color{#0a0}{10n}\!&&\mid\, \color{#c00}m^2 -\, (\color{#0a0}{10n})^2\\ {\rm and}\ \ \ \color{#c00}{2x}\!-\!(\color{#0a0}{-1})\!&&\mid (\color{#c00}{2x})^3\!-(\color{#0a0}{-1})^3\ \ \text{by the Factor Theorem.}\end{eqnarray}$