Question

Show that if P ( x ) is a linear polynomial, then the composition P o P(x) is also linear and has positive slope.

Answer 1

If P(x) is a linear polynomial, then it can be written
\[\large P(x) = mx + b\]

Answer 2

How would I find Find P(P(x))?

Answer 3

Well, to find \(\large P(P(x))\)
replace all instances of x in P(x) with another instance of P(x).

Answer 4

So, y = m(y=mx+b) + b?

Answer 5

y=m(mx+b)+b should suffice.
Now simplify it. :)

Answer 6

y=my+m^2x+mb+b?

Answer 7

Why is there a y?
\[\large P(x) = mx+b\\ \large P(P(x))=m(P(x))+b\\ \large P(P(x))=m(\color{blue}{mx+b})+b\]
Please try simplifying it again.

Answer 8

P=(m^2)x+bm+b

Answer 9

Very good.
\[\large P\circ P(x) = P(P(x))=\color{red}{m^2}x +bm+x\]
And the slope is the coefficient of x (highlighted in red)
Now, what can you say about this new slope?

Answer 10

Sorry a slight correction
\[\large P\circ P(x) = P(P(x))=\color{red}{m^2}x +bm+\color{green}b\]