Question

What is the remainder of the quotient 10x4−91x3+17x2−75x+26÷x−9 ?

Answer 1

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:

1. Calculate \( 9^2 = 81 \)
2. Calculate \( 9^3 = 729 \)
3. 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:

1. \( 10(6561) = 65610 \)
2. \( 91(729) = 66339 \)
3. \( 17(81) = 1377 \)
4. \( 75(9) = 675 \)

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 \(\boxed{-1}\).