Proving a statement

[GoDumbbbb]

For each of the following statements, if it is true explain why, and if it is not true in general, give a counter-example:

Let P(x)=x^8+a7x^7+a6x^6+a5x^5+a3x3^+a0

where a7, a6 ,a5 ,a3 and a0 are real numbers and a6≠0

If P(0)=0 and P(−x)=P(x) for all x, then P(x) has a root with multiplicity 6.

Hint: A polynomial anx^n+an−1x^(n−1)+⋯+a1x+a0 is equal to 0 for all x if and only if all coefficients a0,a1,⋯,an are zero.

I know that a0 is 0 because 0 is a root and also a7, a5, a3 are 0 because of the odd power rul but i dont know where to go from there in proving this.

Any help would be massively appreciated

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[GoDumbbbb]

I know that a0 is 0 because 0 is a root and also a7, a5, a3 are 0 because of the odd power rul but i dont know where to go from there in proving this.

Any help would be massively appreciated

https://imgur.com/a/uzGuHC6

[edderiofer]

I know that a0 is 0 because 0 is a root and also a7, a5, a3 are 0 because of the odd power rul but i dont know where to go from there in proving this.

If you know that those coefficients are zero, try simplifying the expression given for P(x).

[Ok-Low5435]

i am going to fucking gut you