584. Докажите тождество: a) a(b – c) = –a(c – b); б) (a – x)(b + y) + (a – x)(b – y) = 2b(a – x); в) a(b – c + d) = –a(c – b – d); г) (a – b)(c – d) = (b – a)(d – c); д) (a – b)(b – c) + (b – a)(b – c) = 0; e) 2a(a – b) + 4ab = 2a(a + b).

584. Докажите тождество:

а) a(b – c) = –a(c – b)

a(b – c) = ab – ac = -ac + ab = -(ac - ab) = -a(c - b)

Ответ: доказано тождество a(b – c) = –a(c – b)

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б) (a – x)(b + y) + (a – x)(b – y) = 2b(a – x)

(a – x)(b + y) + (a – x)(b – y) = (a - x)((b + y) + (b - y)) = (a - x)(b + y + b - y) = (a - x)2b = 2b(a - x)

Ответ: доказано тождество (a – x)(b + y) + (a – x)(b – y) = 2b(a – x)

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в) a(b – c + d) = –a(c – b – d)

a(b – c + d) = ab - ac + ad = -ac + ab + ad = -(ac - ab - ad) = -a(c - b - d)

Ответ: доказано тождество a(b – c + d) = –a(c – b – d)

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г) (a – b)(c – d) = (b – a)(d – c)

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

Ответ: доказано тождество (a – b)(c – d) = (b – a)(d – c)

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д) (a – b)(b – c) + (b – a)(b – c) = 0

(a – b)(b – c) + (b – a)(b – c) = (a - b)(b - c) - (a - b)(b - c) = 0

Ответ: доказано тождество (a – b)(b – c) + (b – a)(b – c) = 0

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e) 2a(a – b) + 4ab = 2a(a + b).

2a(a – b) + 4ab = 2a² - 2ab + 4ab = 2a² + 2ab = 2a(a + b)

Ответ: доказано тождество 2a(a – b) + 4ab = 2a(a + b)