What is reminder r if polynomial f =(x-1)^10+(x-2)^10 is divided by g =x^2-3x+2?

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You need to use the following rule with respect to polynomial division, such that:

`f(x_1) = r(x_1), f(x_2) = r(x_2),....,f(x_n) = r(x_n)` if polynomial f(x) is divided by polynomial `(x - x_1)(x - x_2)...(x - x_n)`

The problem provides the information that the polynomial `f(x)` is divided by the polynomial `g(x)` , hence, you need to convert the given standard form of `g(x)` into its factored form, such that:

`g(x) = a(x - x_1)(x - x_2)`

You need to evaluate the roots `x_1, x_2` using quadratic formula, such that:

`x_(1,2) = (3+-sqrt(9-8))/2`

`x_(1,2) = (3+-1)/2 => x_1 = 2 ; x_2 = 1`

`g(x) = (x - 1)(x - 2)`

The reminder of division by `g(x) = x^2-3x+2` is `r(x) = ax + b` , hence, using the rule stated above yields:

`{(f(1) = r(1)),(f(2) = r(2)):}`

Evaluating ` f(1)` and `f(2), r(1)` and `r(2)` yields:

`f(1) = (1-1)^10+(1-2)^10 => f(1) = 1`

`f(2) = (2-1)^10+(2-2)^10 => f(2) = 1`

`r(1) = a + b => a + b = 1`

`r(2) = 2a + b => 2a + b = 1`

`a + b = 2a + b => 2a - a = 0 => a = 0 => b = 1`

**Hence, evaluating the reminder ` r(x) = ax + b` , under the given conditon, yields `r(x) = 1` .**

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