To find the remainder of the polynomial \( 10x^4 - 91x^3 + 17x^2 - 75x + 26 \) when divided by \( x - 9 \), we can use the Remainder Theorem. According to this theorem, the remainder of a polynomial \( f(x) \) when divided by \( x - c \) is given by \( f(c) \).
In this case, we have \( c = 9 \). Therefore, we need to evaluate the polynomial at \( x = 9 \).
The polynomial is given by:
\[ f(x) = 10x^4 - 91x^3 + 17x^2 - 75x + 26 \]
Now we calculate \( f(9) \):
\[ f(9) = 10(9^4) - 91(9^3) + 17(9^2) - 75(9) + 26 \]
Calculating powers of 9:
\[ 9^2 = 81, \quad 9^3 = 729, \quad 9^4 = 6561 \]
Substituting these values into the equation:
\[ f(9) = 10(6561) - 91(729) + 17(81) - 75(9) + 26 \]
Calculating each term:
\[ 10 \times 6561 = 65610 \] \[ 91 \times 729 = 66339 \] \[ 17 \times 81 = 1377 \] \[ 75 \times 9 = 675 \]
Now substituting these back into \( f(9) \):
\[ f(9) = 65610 - 66339 + 1377 - 675 + 26 \]
Now, combine the terms step by step:
- \( 65610 - 66339 = -729 \)
- \( -729 + 1377 = 648 \)
- \( 648 - 675 = -27 \)
- \( -27 + 26 = -1 \)
Thus, the remainder when dividing the polynomial \( 10x^4 - 91x^3 + 17x^2 - 75x + 26 \) by \( x - 9 \) is
\[ \boxed{-1}. \]
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