Algebra

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Polynomial Identities
$$(a+b)^2 = a^2 + 2ab + b^2$$

$$(a+b)(c+d) = ac + ad + bc + bd$$

$$a^2 - b^2 = (a+b)(a-b)$$

$$a^3 \pm b^3 = (a\pm b)(a^2 \mp ab + b^2)$$

$$x^2 + (a+b)x + ab = (x + a)(x + b)$$

if $$ax^2 + bx + c = 0$$ then $$x = -b \pm \sqrt{\frac{b^2 - 4ac}{2a}}$$

Powers
$$x^a x^b = x^{(a + b)}$$

$$x^a y^a = (xy)^a $$

$$(x^a)^b = x^{ab}$$

$$x^{-a} = \frac{1}{x^a}$$

$$x^{(a-b)} = \frac{x^a}{x^b}$$

Logarithms
$$y = \log_{10} x$$ if and only if $$x=b^y$$

$$\log_{b} 1 = 0$$

$$\log_{b} b = 1$$

$$\log_{b} (x+y) = \log_{b}x + \log_{b}y$$

$$\log_{b} \frac{x}{y} = \log_{b}x - \log_{b}y$$

$$\log_{b} x^n = n \log_{b} x$$

$$\log_{b} x = \log_{b} c \times \log_{c} x = \log_{c} x / \log_{c} b$$