1) (a+b+c)(a+b-1) [Hint: consider x=a+b]
2) (a+b+c+d)(a+b-c-d) [Hint: consider x=a+b and y=c+d]
3) (x^2+x+1)(x^2-x+1)
4) What can be multiplied to x-2 to give a product containing the term -3x^2?Findthe polynomial. [Note: There are more than one answer.]
數學問題.F.1→F.2
2010-07-30 11:19 pm
回答 (2)
2010-07-30 11:59 pm
✔ 最佳答案
1) Let x = a+b(a+b+c)(a+b-1)
= (x+c)(x-1)
= x^2 + cx - x - c
= (a+b)^2 + c(a+b) - (a+b) - c
= (a+b)(a+b) + ca + cb - a - b - c
= a^2 + ab + ab + b^2 + ca + cb - a - b - c
= a^2 + 2ab + b^2 + ca + cb - a - b - c
2) Let x = a+b and y = c+d
(a+b+c+d)(a+b-c-d)
= (x+y)[x - (c+d)]
= (x+y)(x-y)
= x^2 + xy - xy - y^2
= x^2 - y^2
= (a+b)^2 - (c+d)^2
= (a+b)(a+b) - (c+d)(c+d)
= a^2 + ab + ab + b^2 - (c^2 + cd + cd + d^2)
= a^2 + 2ab + b^2 - c^2 - 2cd - d^2
= a^2 + b^2 - c^2 - d^2 + 2ab - 2cd
3) (x^2+x+1)(x^2-x+1)
= x^4 - x^3 + x^2 + x^3 - x^2 + x + x^2 - x + 1
= x^4 - x^3 + x^3 + x^2 - x^2 + x^2 + x - x + 1
= x^4 + x^2 + 1
4) The polynomial can be, e.g. -3x+1 , 3x^2 /2 + 1
For (-3x+1), (-3x+1)(x-2) = -3x^2 + x + 6x - 2 = -3x^2 + 7x - 2, which contains the term -3x^2
For (3/2 x^2 + 1), (3x^2 /2+ 1)(x-2) = 3x^3 /2 - 3x^2 + x - 2, which contains the term -3x^2
參考: Knowledge is power.
2010-07-31 7:35 am
1) Let x = a+b
(a+b+c)(a+b-1)
= (x+c)(x-1)
= x^2 + cx - x - c
= (a+b)^2 + c(a+b) - (a+b) - c
= (a+b)(a+b) + ca + cb - a - b - c
= a^2 + ab + ab + b^2 + ca + cb - a - b - c
= a^2 + 2ab + b^2 + ca + cb - a - b - c
2) Let x = a+b and y = c+d
(a+b+c+d)(a+b-c-d)
= (x+y)[x - (c+d)]
= (x+y)(x-y)
= x^2 + xy - xy - y^2
= x^2 - y^2
= (a+b)^2 - (c+d)^2
= (a+b)(a+b) - (c+d)(c+d)
= a^2 + ab + ab + b^2 - (c^2 + cd + cd + d^2)
= a^2 + 2ab + b^2 - c^2 - 2cd - d^2
= a^2 + b^2 - c^2 - d^2 + 2ab - 2cd
3) (x^2+x+1)(x^2-x+1)
= x^4 - x^3 + x^2 + x^3 - x^2 + x + x^2 - x + 1
= x^4 - x^3 + x^3 + x^2 - x^2 + x^2 + x - x + 1
= x^4 + x^2 + 1
4) The polynomial can be, e.g. -3x+1 , 3x^2 /2 + 1
For (-3x+1), (-3x+1)(x-2) = -3x^2 + x + 6x - 2 = -3x^2 + 7x - 2, which contains the term -3x^2
For (3/2 x^2 + 1), (3x^2 /2+ 1)(x-2) = 3x^3 /2 - 3x^2 + x - 2, which contains the term -3x^2
(a+b+c)(a+b-1)
= (x+c)(x-1)
= x^2 + cx - x - c
= (a+b)^2 + c(a+b) - (a+b) - c
= (a+b)(a+b) + ca + cb - a - b - c
= a^2 + ab + ab + b^2 + ca + cb - a - b - c
= a^2 + 2ab + b^2 + ca + cb - a - b - c
2) Let x = a+b and y = c+d
(a+b+c+d)(a+b-c-d)
= (x+y)[x - (c+d)]
= (x+y)(x-y)
= x^2 + xy - xy - y^2
= x^2 - y^2
= (a+b)^2 - (c+d)^2
= (a+b)(a+b) - (c+d)(c+d)
= a^2 + ab + ab + b^2 - (c^2 + cd + cd + d^2)
= a^2 + 2ab + b^2 - c^2 - 2cd - d^2
= a^2 + b^2 - c^2 - d^2 + 2ab - 2cd
3) (x^2+x+1)(x^2-x+1)
= x^4 - x^3 + x^2 + x^3 - x^2 + x + x^2 - x + 1
= x^4 - x^3 + x^3 + x^2 - x^2 + x^2 + x - x + 1
= x^4 + x^2 + 1
4) The polynomial can be, e.g. -3x+1 , 3x^2 /2 + 1
For (-3x+1), (-3x+1)(x-2) = -3x^2 + x + 6x - 2 = -3x^2 + 7x - 2, which contains the term -3x^2
For (3/2 x^2 + 1), (3x^2 /2+ 1)(x-2) = 3x^3 /2 - 3x^2 + x - 2, which contains the term -3x^2
參考: 自己
收錄日期: 2021-04-25 17:28:11
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