Utajifunza:
- Bidhaa maalum
- Bidhaa ya Binomials mbili
- Bidhaa ya Jumla na Tofauti ya maneno mawili
- Upanuzi
Kuzidisha kwa aina fulani za semi za aljebra kunaweza kupatikana kiakili kwa kutumia sheria fulani. Kuzidisha vile kunaitwa kama bidhaa maalum .
Bidhaa za Binomials mbili
Wacha tupate bidhaa za \((x+y)(x+z)\) ,
Hapa \((x + y)(x + z) = x(x + z)+y(x + z) = x^2 + xz + yx + yz\)
\(x^2 +x(y + z) + yz\)
Kwa hivyo, \((x + y)(x + z) = x^2 +(y+ z)x + yz\)
Kwa njia hiyo hiyo, tunaweza kupata bidhaa maalum zifuatazo kwa urahisi:
\((x + y)(x - z) = x^2 +(y - z)x - yz\)
\((x - y)(x + z) = x^2 +(z - y)x - yz\)
\((x - y)(x - z) = x^2 - (y+z)x + yz\)
Mifano: Tafuta bidhaa zifuatazo-
- \((2a+3)(2a+4) = (2a)^2 + (3+4)(2a) + (3)(4) = 4a^2 + 14a + 12\)
- \((2m - p^2)(2m + q^2) = (2m)^2 + (q^2 - p^2)(2m) - (q^2)(p^2) = 4m^2 + 2m(q^2 - p^2) - q^2p^2\)
Bidhaa ya Jumla na Tofauti ya maneno mawili
\((x + y)(x - z) = x^2 + (y - z)x -yz\)
badilisha z kwa y
\(⇒ (x + y)(x - y) = x^2 - y^2 \)
Upanuzi
Wakati usemi wa aljebra unazidishwa wenyewe hadi wa pili, tatu, au nguvu nyingine yoyote basi mchakato huo unaitwa upanuzi.
\((x +y)^2 = x^2 + 2xy + y^2 \)
\((x - y)^2 = x^2 - 2xy + y^2\)
\((x + y+ z)^2 = x^2 + y^2 + z^2 + 2(xy+yz+xz)\)
\((x + y)^3 = x^3+y^3+ 3xy(x+y) \)
\((x - y)^3 = x^3 - y^3 - 3xy(x - y)\)
Mifano:
\((\sqrt2 + x)^2 = (\sqrt2)^2 + 2 \cdot \sqrt2.x + x^2 \\ 2+2\sqrt2x+x^2\)
\((104)^2 = (100+4)^2 = (100)^2 + 2⋅ 4 ⋅100+ 42 = 10000 + 800 + 16 =10816 \)
Perfect Square Trinomial
Utatu wowote unaoweza kuonyeshwa kama \( (x^2 + 2xy + y^2) \textrm{ au } (x^2 - 2xy + y^2)\) unajulikana kama utatu kamili wa mraba .
\(x^2 + 2xy + y^2\) ni mraba kamili wa (x+y) na \(x^2 - 2xy + y^2\) ni mraba kamili wa (x−y).
