Question
What is the remainder of the quotient of x4−10x3+12x2+7x−1÷x−1 ?(1 point)
Answers
Answered by
GPT-4o mini
To find the remainder of the polynomial \( x^4 - 10x^3 + 12x^2 + 7x - 1 \) when divided by \( x - 1 \), we can use the Remainder Theorem. According to the Remainder Theorem, the remainder of the division of a polynomial \( f(x) \) by \( x - c \) is equal to \( f(c) \).
In this case, we need to evaluate the polynomial at \( c = 1 \):
\[
f(x) = x^4 - 10x^3 + 12x^2 + 7x - 1
\]
Now we substitute \( x = 1 \):
\[
f(1) = 1^4 - 10(1^3) + 12(1^2) + 7(1) - 1
\]
Calculating each term:
\[
f(1) = 1 - 10 + 12 + 7 - 1
\]
Now combine these values:
\[
f(1) = 1 - 10 + 12 + 7 - 1 = -9 + 19 = 10
\]
Thus, the remainder when \( x^4 - 10x^3 + 12x^2 + 7x - 1 \) is divided by \( x - 1 \) is:
\[
\boxed{10}
\]
In this case, we need to evaluate the polynomial at \( c = 1 \):
\[
f(x) = x^4 - 10x^3 + 12x^2 + 7x - 1
\]
Now we substitute \( x = 1 \):
\[
f(1) = 1^4 - 10(1^3) + 12(1^2) + 7(1) - 1
\]
Calculating each term:
\[
f(1) = 1 - 10 + 12 + 7 - 1
\]
Now combine these values:
\[
f(1) = 1 - 10 + 12 + 7 - 1 = -9 + 19 = 10
\]
Thus, the remainder when \( x^4 - 10x^3 + 12x^2 + 7x - 1 \) is divided by \( x - 1 \) is:
\[
\boxed{10}
\]