To find the remainder of the polynomial division \( 10x^4 - 91x^3 + 17x^2 - 75x + 26 \) divided by \( x - 9 \), we can use the Remainder Theorem. According to the theorem, the remainder of the division of a polynomial \( f(x) \) by \( x - c \) is equal to \( f(c) \).
In this case, we have:
\[ f(x) = 10x^4 - 91x^3 + 17x^2 - 75x + 26 \] and we want to evaluate \( f(9) \).
Substituting \( x = 9 \) into the polynomial:
\[ f(9) = 10(9^4) - 91(9^3) + 17(9^2) - 75(9) + 26 \]
Calculating each term step-by-step:
- Calculate \( 9^2 = 81 \)
- Calculate \( 9^3 = 729 \)
- Calculate \( 9^4 = 6561 \)
Now substituting these values into the polynomial:
\[ f(9) = 10(6561) - 91(729) + 17(81) - 75(9) + 26 \]
Calculating each term:
- \( 10(6561) = 65610 \)
- \( 91(729) = 66339 \)
- \( 17(81) = 1377 \)
- \( 75(9) = 675 \)
Now substituting these back:
\[ f(9) = 65610 - 66339 + 1377 - 675 + 26 \]
Calculating within:
First, combine \( 65610 - 66339 \): \[ 65610 - 66339 = -729 \]
Now, continue to combine: \[ -729 + 1377 = 648 \] \[ 648 - 675 = -27 \] \[ -27 + 26 = -1 \]
Thus, the final result for the calculation is:
\[ f(9) = -1 \]
Therefore, the remainder when \( 10x^4 - 91x^3 + 17x^2 - 75x + 26 \) is divided by \( x - 9 \) is \(\boxed{-1}\).