问题1.使用合适的身份获得以下每个产品。
(i)(x + 3)(x + 3)
解决方案：
(x + 3) (x + 3)
Putting formula (a + b)2 = a2 + b2+ 2ab
Put a = x & b = 3
(x + 3) (x +3 ) = (x + 3)2
= x2 + 6x + 9
(ii)(2年+ 5岁)(2年+ 5岁)
解决方案：
(2y + 5) (2y + 5)
Putting formula (a + b)2 = a2 + b2 + 2ab
Put a = 2y & b = 5
(2y + 5) (2y + 5) = (2y + 5)2
= 4y2 + 20y + 25
(iii)(2a – 7)(2a – 7)
解决方案：
(2a – 7) (2a – 7)
Putting formula (x – y)2 = x2 + y2 – 2xy
Put x = 2a & y = 7
(2a – 7) (2a – 7) = (2a – 7)2
= 4a2 – 28a + 49
(iv)(3a – 1/2)(3a – 1/2)
解决方案：
(3a – 1/2) (3a – 1/2)
Putting formula (x – y)2 = x2 + y2 – 2xy
Put x = 3a & y = 1/2
= (3a)2 + (1/2)2 – 2 (3a) (1/2)
= 32a2 + 1/4 – 3a
= 9a2 + 1/4 – 3a
(v)(1.1m – 0.4)(1.1m + 0.4)
解决方案：
(1.1m – 0.4) (1.1m + 0.4)
Putting formula (a + b) (a – b) = a2– b2
Put a = 1.1m & b = 0.4
(1.1m – 0.4) (1.1m + 0.4) = (1.1m)2 – (0.4)2
= 1.21m2 – 0.16
(vi)(a 2 + b 2 )(– a 2 + b 2 )
解决方案：
(a2 + b2) (– a2 + b2)
(a2 + b2) (– a2 + b2) = (b2 + a2) (b2 – a2)
Putting formula (x + y) (x – y) = x2 – y2
Put x = b2 & y = a2
(b2 + a2) (b2 – a2) = b2×2 – a2×2
= b4 – a4
(vii)(6x – 7)(6x + 7)
解决方案：
(6x – 7) (6x + 7)
Putting formula (a – b) (a + b) = a2 – b2
Put a = 6x & b = 7
(6x – 7) (6x + 7) = (6x)2 – 72
= 36x2 – 49
(viii)(– a + c)(– a + c)
解决方案：
(– a + c) (– a + c)
(– a + c) (– a + c) = (c – a) (c – a)
Putting formula (x – y)2 = x2 + y2 – 2xy
Put x = c & y = a
(c – a) (c – a) = c2 + a2 – 2ca
= a2 + c2 – 2ac
(ix)(x / 2 + 3y / 4)(x / 2 + 3y / 4)
解决方案：
(x/2 + 3y/4) (x/2 + 3y/4)
Putting formula (a + b)2 = a2 + b2 + 2ab
put a = x/2 & b = 3y/4
= (x/2)2 + (3y/4)2 + 2 (x/2) (3y/4)
= x2/4 + 9y2/16 + 3xy/4
(x)(7a – 9b)(7a – 9b)
解决方案：
(7a – 9b) (7a – 9b)
Putting formula (x – y)2 = x2+ y2– 2xy
Put x = 7a & y = 9b
(7a – 9b) (7a – 9b) = (7a)2 + (9b)2 – 2(7a)(9b)
= 49a2 + 81b2 – 126ab
问题2.使用标识(x + a)(x + b)= x 2 +(a + b)x + ab查找以下乘积。
(i)(x + 3)(x + 7)
解决方案：
(x + 3) (x + 7)
Formula (x + a) (x + b) = x2 + (a + b) x + ab
Put a = 3 & b = 7
= x2 + (3 + 7) x + (3 * 7)
= x2 +10x + 21
(ii)(4x + 5)(4x + 1)
解决方案：
(4x + 5) (4x + 1)
Formula (y + a) (y + b) = y2 + (a + b) y + ab
Put y = 4x , a = 5 & b = 1
= (4x)2 + (5 + 1) 4x + (5 * 1)
= 16x2 + 24x + 5
(iii)(4x – 5)(4x – 1)
解决方案：
(4x – 5) (4x – 1)
Formula (y + a) (y + b) = y2 + (a + b) y + ab
Put y = 4x , a = -5 & b = -1
= (4x)2 + (-5 – 1) 4x + (-5 * -1)
= 16x2 – 24x + 5
(iv)(4x + 5)(4x – 1)
解决方案：
(4x + 5) (4x – 1)
Formula (y + a) (y + b) = y2 + (a + b) y + ab
Put y = 4x , a = 5 & b = -1
= (4x)2 + (5 – 1) 4x + (5 * -1)
= 16x2 + 16x – 5
(v)(2x + 5y)(2x + 3y)
解决方案：
(2x + 5y) (2x + 3y)
Formula (t + a) (t + b) = t2 + (a + b) t + ab
Put t = 2x , a = 5y & b = 3y
= (2x)2 + ( 5y + 3y) 2x + (5y * 3y)
= 4x2 + 16xy + 15y2
(vi)(2a 2 + 9)(2a 2 + 5)
解决方案：
(2a2 + 9) (2a2 + 5)
Formula (x + y) (x + z) = x2 + (y + z) x + yz
Put x = 2a2 , y = 9 & z = 5
= (2a2)2 + (9 + 5) 2a2 + (9 * 5)
= 4a4 + 28a2 + 45
(vii)(xyz – 4)(xyz – 2)
解决方案：
(xyz – 4) (xyz – 2)
Formula (t + a) (t + b) = t2 + (a + b) t + ab
Put t = xyz , a = -4 & b = -2
= (xyz)2 + (-4 + (-2)) xyz + ((-4) * (-2))
= x2y2z2 – 6xyz + 8
问题3.使用标识找到以下正方形。
(i)(b – 7) 2
解决方案：
(b – 7)2
Using Formula (x – y)2 = x2 + y2 – 2xy
Putting x = b & y = 7
= b2 + 72 – 2(b)(7)
= b2 – 14b + 49
(ii)(xy + 3z) 2
解决方案：
(xy + 3z)2
Using Formula (a + b)2 = a2 + b2 + 2ab
Putting a = xy & b = 3z
= x2y2 + 6xyz + 9z2
(iii)(6x 2 – 5y) 2
解决方案：
(6x2 – 5y)2
Using Formula (a – b) 2 = a2 + b2 – 2ab
Putting a = 6x2 & b = 5y
= 36x4 – 60x2y + 25y2
(iv)[(2m / 3)+(3n / 2)] 2
解决方案：
[(2m/3) + (3n/2)]2
Using Formula (a + b)2 = a2 + b2 + 2ab
Putting a = 2m/3 & b = 3n/2
= (2m/3)2 + (3n/2)2 + 2 (2m/3) (3n/2)
= (4m2/9) + (9n2/4) + 2mn
(v)(0.4p – 0.5q) 2
解决方案：
(0.4p – 0.5q)2
Using Formula (a – b)2 = a2 + b2 – 2ab
Putting a = 0.4p & b = 0.5q
= 0.16p2 – 0.4pq + 0.25q2
(vi)(2xy + 5y) 2
解决方案：
(2xy + 5y)2
Using Formula (a + b)2 = a2 + b2 + 2ab
Putting a = 2xy & b = 5y
= (2xy)2 + (5y)2 + 2 (2xy) (5y)
= 4x2y2 + 20xy2 + 25y2
问题4.简化。
(i)(a 2 – b 2 ) 2
解决方案：
(a2 – b2)2
Putting formula (x – y)2 = x2 + y2 – 2xy
Put x = a2 & y = b2
= a4 + b4 – 2a2b2
(ii)(2x + 5) 2 –(2x – 5) 2
解决方案：
(2x + 5)2 – (2x – 5)2
Putting formula (a + b)2 = a2 + b2+ 2ab & (a – b)2 = a2 + b2 – 2ab
= 4x2 + 20x + 25 – (4x2 – 20x + 25)
= 4x2 + 20x + 25 – 4x2 + 20x – 25
= 40x
(iii)(7m – 8n) 2 +(7m + 8n) 2
解决方案：
(7m – 8n)2 + (7m + 8n)2
Putting formula (a – b)2 = a2 + b2 – 2ab & (a + b)2 = a2 + b2+ 2ab
= (49m2 – 112mn + 64n2 ) + (49m2 + 112mn + 49n2)
= 98m2 + 128n2
(iv)(4m + 5n) 2 +(5m + 4n) 2
解决方案：
(4m + 5n)2 + (5m + 4n)2
Putting formula (a + b)2 = a2 + b2+ 2ab
= (16m2 + 40mn + 25n2) + (25m2 + 40mn + 16n2)
= 41m2 + 80mn + 41n2
(v)(2.5p – 1.5q) 2 –(1.5p – 2.5q) 2
解决方案：
(2.5p – 1.5q)2 – (1.5p – 2.5q)2
Putting formula (a – b)2 = a2 + b2 – 2ab
= (6.25p2 – 7.5pq + 2.25q2) – (2.25p2 + 7.5pq – 6.25q2)
= 4p2 – 4q2
(vi)(ab + bc) 2 – 2ab 2 c
解决方案：
(ab + bc)2 – 2ab²c
Putting formula (a + b)2 = a2 + b2 + 2ab
= (a2b2 + 2ab2c + b2c2) – 2ab2c
= a2b2 + b2c2
(vii)(m 2 – n 2 m) 2 + 2m 3 n 2
解决方案：
(m2 – n2m)2 + 2m3n2
Putting formula (a – b)2 = a2 + b2 – 2ab
= (m4 – 2m3n2 + m2n4) + 2m3n2
= m4 + m2n4