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The problem asks for the remainder when dividing the polynomial \( x^4 - 10x^3 + 12x^2 + 7x - 1 \) by \( x - 1 \).

To solve this, we can use **synthetic division** or **the Remainder Theorem**.

### Remainder Theorem:
According to the remainder theorem, when dividing a polynomial \( f(x) \) by \( x - c \), the remainder is \( f(c) \).

Here, the divisor is \( x - 1 \), so \( c = 1 \). We simply evaluate the polynomial at \( x = 1 \):

\[
f(x) = x^4 - 10x^3 + 12x^2 + 7x - 1
\]

Substituting \( x = 1 \):

\[
f(1) = (1)^4 - 10(1)^3 + 12(1)^2 + 7(1) - 1
\]
\[
f(1) = 1 - 10 + 12 + 7 - 1 = 9
\]

Thus, the remainder is **9**.

Would you like a detailed explanation of synthetic division as well? Here are some follow-up questions:

1. How do you perform synthetic division for this problem?
2. What does the remainder theorem tell us about polynomial division?
3. Can you always use the remainder theorem for any divisor?
4. What happens when the divisor is more complicated (e.g., not in the form \( x - c \))?
5. How would the quotient look after division in this case?

**Tip:** The remainder theorem is a quick and powerful tool for checking divisibility and remainders in polynomial division.

What is the remainder of the quotient of x^4 - 10x^3 + 12x^2 + 7x - 1 divided by x - 1?

Find the remainder of the polynomial division of x^4 - 10x^3 + 12x^2 + 7x - 1 by x - 1 using the Remainder Theorem. Learn step-by-step how to apply this theorem in algebra. Suitable for high school students, Grades 10-12.

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