Question

What is the remainder of the quotient 10x^4−91x^3+17x^2−75x+26÷x−9 ?(1 point)

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. The Remainder Theorem states that the remainder of the division of a polynomial \(f(x)\) by \(x - c\) is equal to \(f(c)\).

Here, our polynomial \(f(x) = 10x^4 - 91x^3 + 17x^2 - 75x + 26\) and we need to evaluate it at \(c = 9\).

Calculating \(f(9)\):

\[
f(9) = 10(9^4) - 91(9^3) + 17(9^2) - 75(9) + 26
\]

Calculating each term step-by-step:

1. \(9^2 = 81\)
2. \(9^3 = 729\)
3. \(9^4 = 6561\)

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 back:

\[
f(9) = 65610 - 66339 + 1377 - 675 + 26
\]

Calculating step-by-step:

1. \(65610 - 66339 = -729\)
2. \(-729 + 1377 = 648\)
3. \(648 - 675 = -27\)
4. \(-27 + 26 = -1\)

Thus, the remainder when \(10x^4 - 91x^3 + 17x^2 - 75x + 26\) is divided by \(x - 9\) is \(-1\).

So, the final answer is:

\[
\boxed{-1}
\]