作业帮高次方程零点 与转折点的关系
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高次方程零点 与转折点的关系
1The graph of every polynomial function of degree n has at most n-1 turning points
2If a polynomial function has n distinct real zeros,then its graph has exactly n-1 turning points
不太理解 中文意思知道 为什么第二条是exactly啊
1The graph of every polynomial function of degree n has at most n-1 turning points
2If a polynomial function has n distinct real zeros,then its graph has exactly n-1 turning points
不太理解 中文意思知道 为什么第二条是exactly啊
The zeros of the function are the values of x that would make the function equal 0.
An nth degree polynomial in one variable has at most n real zeros.There are exactly n real or complex zeros.
eg: find zeros of y = x^3-4x^2+25x-100 ==> (x-4)(x^2 +25) = 0 ==>
x-4=0 and x^2 +25= 0 ==> x= 4 and x^2 = -25
if in a real number system, we cannot take square root of negative, but in a complex number system we can do it, so x = +/- 5i. for this one, we have three zeros : x = 4, x = 5i and x = -5i (x=4 real and 5i/-5i complex )
Also an nth degree polynomial in one variable has at most n-1 relative extrema (relative maximums or relative minimums). Since a relative extremum is a turn in the graph, you could also say there are at most n-1 turns (turning points)
for the second one, it metioned that "polynomial function has n distinct real zeros", which means no complex numbers exist, so it could get max turning points(exactly n-1)
eg: y = x^3-x^2-6x y=x(x-3)(x+2) all zeros are real numbers, so this function has 3 relative extrema, resulting in 2 turning points
再问： 能给我中文的解释么 。。。。。。。。
再答： turning point 其实就相当于极值 一个n次幂（最高次）的函数最多有n-1个极值==》极值的求法涉及到求导（导数）不知道你学过没有？求polynomial function 的导，实际上就是降幂求根【在polynomial function中，最高次幂是几 就有几根个，根可能是实数也可能是复数 导数的实数根就是原函数的极值】 例如 最高次幂是5，降幂后变为4 最多有4个实数根 对于原函数来说就是最多5-1个极值 即turning point 复数--例一中的-5i 和5i 就是复数根 第二条中明确说明了 "n distinct real zeros" ，就是说没有复数根 所以极值全部为实数 个数最大 即有且只有n-1个 中心思想就是 根=实数根+复数根 当复数根=0 所有根都为实数
An nth degree polynomial in one variable has at most n real zeros.There are exactly n real or complex zeros.
eg: find zeros of y = x^3-4x^2+25x-100 ==> (x-4)(x^2 +25) = 0 ==>
x-4=0 and x^2 +25= 0 ==> x= 4 and x^2 = -25
if in a real number system, we cannot take square root of negative, but in a complex number system we can do it, so x = +/- 5i. for this one, we have three zeros : x = 4, x = 5i and x = -5i (x=4 real and 5i/-5i complex )
Also an nth degree polynomial in one variable has at most n-1 relative extrema (relative maximums or relative minimums). Since a relative extremum is a turn in the graph, you could also say there are at most n-1 turns (turning points)
for the second one, it metioned that "polynomial function has n distinct real zeros", which means no complex numbers exist, so it could get max turning points(exactly n-1)
eg: y = x^3-x^2-6x y=x(x-3)(x+2) all zeros are real numbers, so this function has 3 relative extrema, resulting in 2 turning points
再问： 能给我中文的解释么 。。。。。。。。
再答： turning point 其实就相当于极值 一个n次幂（最高次）的函数最多有n-1个极值==》极值的求法涉及到求导（导数）不知道你学过没有？求polynomial function 的导，实际上就是降幂求根【在polynomial function中，最高次幂是几 就有几根个，根可能是实数也可能是复数 导数的实数根就是原函数的极值】 例如 最高次幂是5，降幂后变为4 最多有4个实数根 对于原函数来说就是最多5-1个极值 即turning point 复数--例一中的-5i 和5i 就是复数根 第二条中明确说明了 "n distinct real zeros" ，就是说没有复数根 所以极值全部为实数 个数最大 即有且只有n-1个 中心思想就是 根=实数根+复数根 当复数根=0 所有根都为实数