为证:(x的平方-4)(x-3)(x-7)+100的值为非负数.
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为证:(x的平方-4)(x-3)(x-7)+100的值为非负数.
为证:(x的平方-4)(x-3)(x-7)+100的值为非负数.
为证:(x的平方-4)(x-3)(x-7)+100的值为非负数.
(x^2 - 4)(x - 3)(x - 7) + 100
= (x - 2)(x + 2)(x - 3)(x - 7) + 100
= [(x - 2)(x - 3)][(x + 2)(x - 7)] + 100
= (x^2 - 5x + 6)(x^2 - 5x - 14) + 100
= (x^2 - 5x)(x^2 - 5x - 14) + 6(x^2 - 5x - 14) + 100
= (x^2 - 5x)(x^2 - 5x) - 14(x^2 - 5x) + 6x^2 - 30x - 84 + 100
= (x^2 - 5x)^2 - 14x^2 + 70x + 6x^2 - 30x + 16
= (x^2 - 5x)^2 - 8x^2 + 40x + 16
= (x^2 - 5x)^2 - 8(x^2 - 5x) + 4^2
= [(x^2 - 5x) - 4]^2
= (x^2 - 5x - 4)^2
≥ 0
(x^2 - 4)(x - 3)(x - 7) + 100
= (x - 2)(x + 2)(x - 3)(x - 7) + 100
= [(x - 2)(x - 3)][(x + 2)(x - 7)] + 100
= (x^2 - 5x + 6)(x^2 - 5x - 14) + 100
=(x^2 - 5x)^2- 8(x^2 - 5x) - 84 + 100
=(x^2 - 5x)^2- 8(x^2 - 5x) +16
= (x^2 - 5x - 4)^2
≥ 0
式中把x^2 - 5x看成一个整体