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 the division of a polynomial \( f(x) \) by \( x - c \) is equal to \( f(c) \).
Here, our polynomial is:
\[ f(x) = 10x^4 - 91x^3 + 17x^2 - 75x + 26 \]
And we need to evaluate this polynomial at \( x = 9 \):
\[ f(9) = 10(9)^4 - 91(9)^3 + 17(9)^2 - 75(9) + 26 \]
Calculating each term step-by-step:
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Calculate \( 9^2 \): \[ 9^2 = 81 \]
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Calculate \( 9^3 \): \[ 9^3 = 729 \]
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Calculate \( 9^4 \): \[ 9^4 = 6561 \]
Now substituting these values back into \( f(9) \):
\[ f(9) = 10(6561) - 91(729) + 17(81) - 75(9) + 26 \]
Calculating each multiplication:
- \( 10(6561) = 65610 \)
- \( 91(729) = 66339 \)
- \( 17(81) = 1377 \)
- \( 75(9) = 675 \)
Now substituting these results into \( f(9) \):
\[ f(9) = 65610 - 66339 + 1377 - 675 + 26 \]
Now we perform the additions and subtractions step-by-step:
\[ = 65610 - 66339 = -729 \] \[ -729 + 1377 = 648 \] \[ 648 - 675 = -27 \] \[ -27 + 26 = -1 \]
Thus, the remainder when \( 10x^4 - 91x^3 + 17x^2 - 75x + 26 \) is divided by \( x - 9 \) is:
\[ \boxed{-1} \]