Subject: polynomial functions
Who is asking: Student
- Without fully factoring the following show that they all have the same zeros:
- When P(x)=x3-3x2+5x+1 and G(x)=x3-2x2-x+10 are each divided by(x-a) the remainders are equal. At what coordinate point does the graph of P(x) intersect G(x)?
- I organize my thoughts here with a picture.
I 'see' the graphs and the question says they have cross the x-axis at
the same points. One way this could happen would be if they were
multiples of one another:
f(a) = 0 if and only if c f(x) = 0 (c not 0).
Now look again at these expressions, I, II, III, IV.
II = (-1) I.
III = (-4) I
etc. No factoring of anything. In fact, this principle does apply
even if the expression does not factor. There is a big fancy theory
of higher algebra, about polynomials, that says something of this
sort - provided you check real and complex solutions to f(a) = 0.
If you want to impress you teacher, refer to Hilbert's Nullstellensatz!
- Again, I see a picture. They are asking for (at least) one point where
the graphs cross. Let that point be c .
At that point, P(c) = G(c).
Is it possible that they have told you, in disguise, that
P(a) = G(a)? [After all, there is not much else to work with.]
Well yes they have. The division stuff says:
P(x) = (x-a)Q(x) + R1, G(x) = (x-a) H(x) + R2 and R1=R2.
Check out what happens when you substitute x=a !
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