To determine the remainder when [tex]\( p(x) = x^3 - ax^2 + x \)[/tex] is divided by [tex]\( x - a \)[/tex], we can use the Remainder Theorem. The Remainder Theorem states that for a polynomial [tex]\( p(x) \)[/tex], the remainder of the division of [tex]\( p(x) \)[/tex] by [tex]\( x - a \)[/tex] is [tex]\( p(a) \)[/tex].
In this case, [tex]\( p(x) \)[/tex] is given by the polynomial [tex]\( x^3 - ax^2 + x \)[/tex].
Following the steps to find the remainder:
1. We need to evaluate the polynomial [tex]\( p(x) \)[/tex] at [tex]\( x = a \)[/tex]. Thus, we calculate [tex]\( p(a) \)[/tex].
[tex]\[ p(a) = (a)^3 - a(a)^2 + (a) \][/tex]
2. Substitute [tex]\( x = a \)[/tex] into the polynomial:
[tex]\[ p(a) = a^3 - a \cdot a^2 + a \][/tex]
3. Simplify the expression:
[tex]\[ p(a) = a^3 - a^3 + a \][/tex]
[tex]\[ p(a) = 0 + a \][/tex]
[tex]\[ p(a) = a \][/tex]
Thus, the remainder when [tex]\( p(x) = x^3 - ax^2 + x \)[/tex] is divided by [tex]\( x - a \)[/tex] is [tex]\( a \)[/tex].
Therefore, the correct answer is:
[tex]\[ \text{b) } a \][/tex]
