# Factor the polynomial function to determine the x intercepts and the y intercept - y = x^5 - 21x^3 + 80x Enter the x intercepts in order from smallest to largest, then the y intercept.

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### 1 Answer

Given that, `y = x^5 - 21x^3 + 80x`

`=x(x^4-21x^2+80)`

`=x(x^4-16x^2-5x^2+80)`

`=x{x^2(x^2-16)-5(x^2-16)}`

`=x(x^2-16)(x^2-5)`

`=x(x^2-4^2)(x^2-(sqrt5)^2)`

`=x(x-4)(x+4)(x-sqrt5)(x+sqrt5)`

For obtaining x-intercept, put y=0.

So, `x(x-4)(x+4)(x-sqrt5)(x+sqrt5)=0`

`rArr x=0,4,-4,sqrt5, and (–sqrt5).`

Arranging from smallest to largest,

`x=-4, (-sqrt5), 0, sqrt5, and 4.`

For obtaining y-intercept, put x=0.

Then `y=0(0-4)(0+4)(0-sqrt5)(0+sqrt5)=0`

Therefore, the x intercepts are` -4, (-sqrt5), 0, sqrt5, and 4` , and the y intercept is `0` .

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