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1. 若 $3x^2 - x = 1$, 则 $6x^3 + 7x^2 - 5x + 2016$ 的值是多少?
解答: $$6x^3 + 7x^2 - 5x + 2016 = 2x(3x^2 - x - 1) + 9x^2 - 3x + 2016$$ $$= 9x^2 - 3x + 2016 = 3(3x^2 - x - 1) + 2019 = 2019.$$
2. 多项式 $2x^2 - 4xy + 5y^2 - 12y + 13$ 的最小值是多少?
解答: $$2x^2 - 4xy + 5y^2 - 12y + 13 = 2(x^2 - 2xy + y^2) + 3(y^2 - 4y + 4)$$ $$= 2(x-y)^2 + 3(y - 2)^2 + 1 ge 1.$$ 当且仅当 $x = y = 2$ 时取等号.
因此该多项式的最小值为 $1$.
3. 若 $x + y + z = a$, $xy + yz + zx = b$, $xyz = c$, 用 $a, b, c$ 表示 $xy^2 + x^2y + yz^2 + y^2z + z^2x + x^2z$.
解答: $$acdot b = (x^2y + xyz + x^2z) + (xy^2 + y^2z + xyz) + (xyz + yz^2 + xz^2)$$ $$= xy^2 + x^2y + yz^2 + y^2z + z^2x + x^2z + 3xyz$$ $$Rightarrow xy^2 + x^2y + yz^2 + y^2z + z^2x + x^2z = ab - 3c.$$
4. 方程 $(x^2 + 3x - 4)^2 + (2x^2 - 7x + 6)^2 = (3x^2 - 4x + 2)^2$ 的解是多少?
解答:
注意到 $(x^2 + 3x - 4) + (2x^2 - 7x + 6) = 3x^2 - 4x + 2$, 因此考虑换元.
令 $x^2 + 3x - 4 = m$, $2x^2 - 7x + 6 = n$, 则 $$m^2 + n^2 = (m + n)^2 Rightarrow mn = 0.$$ 由此可得 $$x^2 + 3x - 4 = 0Rightarrow (x+4)(x -1) = 0 Rightarrow x_1 = -4, x_2 = 1,$$ 或者 $$2x^2 - 7x + 6 = 0 Rightarrow (2x - 3)(x - 2) = 0 Rightarrow x_3 = {3over2}, x_4 = 2.$$ 综上, 原方程的解为: $$x_1 = -4, x_2 = 1, x_3 = {3over2}, x_4 = 2.$$
5. 设 $x + y = 1$, $x^2 + y^2 = 2$, 求 $x^7 + y^7$ 的值.
解答: $$xy = {1over2}left[(x+y)^2 - (x^2 + y^2) ight] = -{1over2}$$ $$Rightarrow x^3 + y^3 = (x+y)(x^2 - xy + y^2) = {5over2},$$ $$x^4 + y^4 = (x^2 + y^2)^2 - 2x^2y^2 = {7over2}$$ $$Rightarrow x^7 + y^7 = (x^3 + y^3)(x^4 + y^4) - x^3y^3(x + y) = {71over 8}.$$
6. 已知 $m, n$ 都是自然数, 且 $m e n$. 求证: 自然数 $m^4 + 4n^4$ 一定可以表示为四个自然数的平方和.
解答: $$m^4 + 4n^4 = (m^2 + 2n^2)^2 - 4m^2n^2$$ $$= (m^2 + 2n^2 + 2mn)(m^2 + 2n^2 - 2mn)$$ $$= left[(m+n)^2 + n^2 ight]left[(m-n)^2 + n^2 ight]$$ $$= (m+n)^2(m-n)^2 + n^2(m+n)^2 + (m-n)^2n^2 + n^4$$ $$= (m^2 - n^2)^2 + (mn + n^2)^2 + (mn - n^2)^2 + (n^2)^2.$$