Question

What is the remainder of the quotient 10x4−91x3+17x2−75x+26÷x−9
10
𝑥
4
−
91
𝑥
3
+
17
𝑥
2
−
75
𝑥
+
26
÷
𝑥
−
9
?

Answer 1

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:

1. Calculate \( 9^2 \):
\[
9^2 = 81
\]

2. Calculate \( 9^3 \):
\[
9^3 = 729
\]

3. 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:

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

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:

\[
\boxed{-1}
\]