式の展開と因数分解の公式

基本法則

  1. 交換法則
    $A + B = B + A$,$AB = BA$
  2. $(A + B) + C = A + (B + C)$,$(AB)C = A(BC)$
  3. 分配法則
    $A(B + C) = AB + AC$,$(A + B)C = AC + BC$

法則と公式って何が違うの?

記事が出来たのでこちらをドウゾ。

指数法則

$m,nが正の整数であるとき,$

  1. $a^m \times a^n = a^{m+n}$
  2. $(a^m)^n = a^{mn}$
  3. $(ab)^n = a^n b^n$

乗法公式

  1. $m(a + b) = ma + mb$
  2. $(a + b)^2 = a^2 + 2ab + b^2$
    $(a – b)^2 = a^2 – 2ab + b^2$
  3. $(a + b)(a – b) = a^2 – b^2$
  4. $(x + a)(x + b) = x^2 + (a + b)x + ab$
  5. $(ax + b)(cx + d) = acx^2 + (ad + bc)x + bd$
  6. $(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$
    $(a – b)^3 = a^3 – 3a^2 b + 3ab^2 – b^3$
  7. $(a + b + c)^2$
    $= a^2 + b^2 + c^2 + 2ab + 2bc + 2ca$
  8. $(x + a)(x + b)(x + c)$
    $= x^3 + (a + b + c)x^2 + (ab + bc + ca)x + abc$

以下の乗法公式は因数分解に用いることの方が多いぞ。

  1. $(a + b)(a^2 – ab + b^2) = a^3 + b^3$
    $(a – b)(a^2 + ab + b^2) = a^3 – b^3$
  2. $(a + b + c)(a^2 + b^2 + c^2 – ab – bc – ca)$
  3. $= a^3 + b^3 + c^3 – 3abc$

因数分解の公式

  1. $ma + mb = m(a + b)$
  2. $a^2 + 2ab + b^2 =(a + b)^2$
    $a^2 – 2ab + b^2 = (a – b)^2$
  3. $a^2 – b^2 = (a + b)(a – b)$
  4. $x^2 + (a + b)x + ab = (x + a)(x + b)$
  5. $acx^2 + (ad + bc)x + bd = (ax + b)(cx + d)$
  6. $a^3 + 3a^2b + 3ab^2 + b^3 = (a + b)^3$
    $a^3 – 3a^2 b + 3ab^2 – b^3 = (a – b)^3$
  7. $a^2 + b^2 + c^2 + 2ab + 2bc + 2ca$
    $= (a + b + c)^2$
  8. $x^3 + (a + b + c)x^2 + (ab + bc + ca)x + abc$
    $= (x + a)(x + b)(x + c)$
  9. $a^3 + b^3 = (a + b)(a^2 – ab + b^2)$
    $a^3 – b^3 = (a – b)(a^2 + ab + b^2)$
  10. $a^3 + b^3 + c^3 – 3abc$
    $= (a + b + c)(a^2 + b^2 + c^2 – ab – bc – ca)$
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