Pemfaktoran Bentuk x2 + 2xy + y2 dan x2 – 2xy + y2
Dengan menggunkan sifat distributif maka untuk memfaktorkan bentuk aljabar x2 + 2xy + y2 perhatikan uraian berikut.
x2 + 2xy + y2
= x2 + xy + xy + y2
= (x2 + xy) + (xy + y2)
= x(x + y) + y(x + y)
= (x + y)(x + y)
= (x + y)2
Untuk memfaktorkan bentuk aljabar x2 – 2xy + y2 perhatikan uraian berikut.
x2 - 2xy + y2
= x2 - xy - xy + y2
= (x2 - xy) - (xy - y2)
= x(x - y) - y(x - y)
= (x - y)(x - y)
= (x - y)2
Berdasarkan uraian di atas, dapat disimpulkan sebagai berikut.
x2 + 2xy + y2 = (x + y)(x + y) = (x + y)2
x2 - 2xy + y2 = (x - y)(x - y) = (x - y)2
Contoh soal
Faktorkanlah bentuk-bentuk aljabar berikut
1. x2 – 6x + 9
2. p2 – 18p + 81
3. b2 + 6b + 9
4. p2 – 4p + 4
5. x2 – 8x + 16
6. m2 + 2m + 1
Penyelesaian:
1. x2 – 6x + 9
= x2 – 3x – 3x + 9
= (x2 – 3x) – (3x – 9)
= x (x – 3) – 3 (x – 3)
= (x – 3)(x – 3)
= (x – 3)2
2. p2 - 18p + 81
= p2 - 9b - 9p + 81
= (p2 - 9p) - (9p - 81)
= p (p - 9) - 9 (p - 9)
= (p - 9)(p - 9)
= (p - 9)2
3. b2 + 6b + 9
= b2 + 3b + 3b + 9
= (b2 + 3b) + (3b + 9)
= b (b + 3) + 3 (b + 3)
= (b + 3)(b + 3)
= (b + 3)2
4. p2 – 4p + 4
= p2 – 2p – 2p + 4
= (p2 – 2p) – (2p – 4)
= p (p – 2) – 3 (p – 2)
= (p – 2)(p – 2)
= (p – 3)2
5. x2 – 8x + 16
= x2 – 4x – 4x + 16
= (x2 – 4x) – (4x – 16)
= x (x – 4) – 4 (x – 4)
= (x – 4)(x – 4)
= (x – 4)2
6. m2 + 2m + 1
= m2 + m + m + 1
= (m2 + m) + (m + 1)
= m (m + 1) + 1 (m + 1)
= (m + 1)( m + 1)
= (m + 1)2
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