Question

求证:多项式(x2-4﹚﹙x2-10x+21﹚+100的值一定是非负数

Answer 1

证明：

(x²-4﹚﹙x²-10x+21﹚+100

=(x+2)(x-2)(x-3)(x-7)+100

=(x²-5x-14)(x²-5x+6)+100

=(x²-5x)²+6(x²-5x)-14(x²-5x)-84+100

=(x²-5x)²-8(x²-5x)+16

=(x²-5x-4)²

因为(x²-5x-4)²是一非负数,所以多项式(x2-4﹚﹙x2-10x+21﹚+100的值一定是非负数.

愿对你有所帮助!