Since

\(\displaystyle \frac{x^k - y^k}{x-y} = x^{k-1}y^0 + x^{k-2}y^1 + x^{k-3}y^2 + ... + x^1y^{k-2} + y^{k-1}\)

\(\displaystyle {x^k - y^k} = (x-y)(x^{k-1}y^0 + x^{k-2}y^1 + x^{k-3}y^2 + ... + x^1y^{k-2} + y^{k-1})\)

hence \(\displaystyle {x^k - y^k} = (x-y)q_k(x)\)

As usual, we write our polynomial \(\displaystyle p\) as

\(\displaystyle p(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0 \)

\(\displaystyle p(x)-p(y) = \)

\(\displaystyle a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0 \)

\(\displaystyle \\ -(a_ny^n + a_{n-1}y^{n-1} + ... + a_1y + a_0)\)

\(\displaystyle = a_n(x^n-y^n) + a_{n-1}(x^{n-1}-y{n-1}^) + ... + a_1(x-y)\)

The terms have the form \(\displaystyle a_k(x^k-y^k)\). But \(\displaystyle x^k-y^k = (x-y)q_k(x)\), and if we substitute this in we get:

\(\displaystyle p(x)-p(y)=\)

\(\displaystyle a_n(x-y)q_n(x) + a_{n-1}(x-y)q_{n-1}(x) + ... + a_1(x-y)\)

\(\displaystyle (x-y)(a_nq_n(x) + a_{n-1}q_{n-1}(x) + ... a_1)\)

\(\displaystyle = (x-y)q(x)\) where \(\displaystyle q(x) = a_nq_n(x) + a_{n-1}q_{n-1}(x) + ... a_1\)

If \(\displaystyle p(a) = 0\) where \(\displaystyle y = a\) then

\(\displaystyle p(x)-p(a) = (x-a)q(x) \)

Q.E.D

My question is why we don't write \(\displaystyle p(x) - p(y)\) as

\(\displaystyle a_n(x-y)q_n(x) + a_{n-1}(x-y)q_{n-1}(x) + ... + a_1(x-y)q_1(x)\)

\(\displaystyle (x-y)(a_nq_n(x) + a_{n-1}q_{n-1}(x) + ... a_1q_1(x))\).

Can anyone please explain this to me thanks!

\(\displaystyle \frac{x^k - y^k}{x-y} = x^{k-1}y^0 + x^{k-2}y^1 + x^{k-3}y^2 + ... + x^1y^{k-2} + y^{k-1}\)

\(\displaystyle {x^k - y^k} = (x-y)(x^{k-1}y^0 + x^{k-2}y^1 + x^{k-3}y^2 + ... + x^1y^{k-2} + y^{k-1})\)

hence \(\displaystyle {x^k - y^k} = (x-y)q_k(x)\)

As usual, we write our polynomial \(\displaystyle p\) as

\(\displaystyle p(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0 \)

\(\displaystyle p(x)-p(y) = \)

\(\displaystyle a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0 \)

\(\displaystyle \\ -(a_ny^n + a_{n-1}y^{n-1} + ... + a_1y + a_0)\)

\(\displaystyle = a_n(x^n-y^n) + a_{n-1}(x^{n-1}-y{n-1}^) + ... + a_1(x-y)\)

The terms have the form \(\displaystyle a_k(x^k-y^k)\). But \(\displaystyle x^k-y^k = (x-y)q_k(x)\), and if we substitute this in we get:

\(\displaystyle p(x)-p(y)=\)

\(\displaystyle a_n(x-y)q_n(x) + a_{n-1}(x-y)q_{n-1}(x) + ... + a_1(x-y)\)

\(\displaystyle (x-y)(a_nq_n(x) + a_{n-1}q_{n-1}(x) + ... a_1)\)

\(\displaystyle = (x-y)q(x)\) where \(\displaystyle q(x) = a_nq_n(x) + a_{n-1}q_{n-1}(x) + ... a_1\)

If \(\displaystyle p(a) = 0\) where \(\displaystyle y = a\) then

\(\displaystyle p(x)-p(a) = (x-a)q(x) \)

Q.E.D

My question is why we don't write \(\displaystyle p(x) - p(y)\) as

\(\displaystyle a_n(x-y)q_n(x) + a_{n-1}(x-y)q_{n-1}(x) + ... + a_1(x-y)q_1(x)\)

\(\displaystyle (x-y)(a_nq_n(x) + a_{n-1}q_{n-1}(x) + ... a_1q_1(x))\).

Can anyone please explain this to me thanks!

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