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Homework Statement
A polynomial [tex]P(x)=(x-b)^7Q(x)[/tex]
a) Show that [tex]P(b)=P ' (b)=0[/tex]
b) Hence find a and b, if [tex](x-1)^7[/tex] is a factor of: [tex]P(x)=x^7+3x^6+ax^5+x^4+3x^3+bx^2-x-1[/tex]
Homework Equations
If [tex]P(x)=Q(x)R(x)[/tex]
Then [tex]P ' (x)=Q ' (x)R(x)+Q(x)R ' (x)[/tex]
I can't think of anything for the factoring aspect of the question.
The Attempt at a Solution
For a)
[tex]P(b)=(b-b)^7Q(b)=0[/tex]
[tex]P ' (x)=7(x-b)^6Q(x)+(x-b)^7Q'(x)[/tex]
[tex]P'(b)=7(b-b)^6Q(b)+(b-b)^7Q'(b)=0[/tex]
But for b) I have no idea how to apply anything from a) to answer the question. Any ideas?