If x = 1 is a root of the equation x^4 - 9 x^3 + 20 x^2 - 12x=0, then the sum of the other roots of the equation equals:

Options: 

4

-2

8

16

-16

-4

-8

12

2


Given the polynomial equation \(x^4 - 9x^3 + 20x^2 - 12x = 0\), we know that \(x = 1\) is a root.

First, we can factor \(x = 1\) out of the polynomial. If \(x = 1\) is a root, \((x - 1)\) is a factor of the polynomial.

We can perform polynomial division or synthetic division to factor out \((x - 1)\).

Using synthetic division:
1. Write down the coefficients of the polynomial: \(1, -9, 20, -12, 0\).
2. Use \(x = 1\) as the root.

\[
\begin{array}{r|rrrrr}
 1 & 1 & -9 & 20 & -12 & 0 \\
   &   & 1  & -8 & 12  & 0 \\
\hline
   & 1 & -8 & 12 & 0  & 0 \\
\end{array}
\]

The quotient is \(x^3 - 8x^2 + 12x\).

Now, we have the polynomial factored as:
\[
(x - 1)(x^3 - 8x^2 + 12x) = 0
\]

We can further factor \(x^3 - 8x^2 + 12x\):
\[
x^3 - 8x^2 + 12x = x(x^2 - 8x + 12)
\]

Factoring \(x^2 - 8x + 12\):
\[
x^2 - 8x + 12 = (x - 6)(x - 2)
\]

Thus, the polynomial can be written as:
\[
(x - 1)x(x - 6)(x - 2) = 0
\]

The roots are \(x = 1\), \(x = 0\), \(x = 6\), and \(x = 2\).

The sum of the other roots (excluding \(x = 1\)) is:
\[
0 + 6 + 2 = 8
\]

Therefore, the sum of the other roots is:
\[
\boxed{8}
\]

Do you want details or have any questions? Here are five questions you can ask:

1. How do you perform synthetic division for polynomials?
2. Can you explain the factorization process for cubic and quadratic polynomials?
3. How do you determine the roots of a polynomial equation using factoring?
4. What is the relationship between the coefficients and the roots of a polynomial?
5. How can polynomial division be used to simplify solving higher degree equations?

**Tip:** Always check your factored forms by expanding them back to the original polynomial to ensure correctness.

Explore how to find the sum of roots in the polynomial equation x^4 - 9x^3 + 20x^2 - 12x = 0, where x = 1 is a known root. Learn about polynomial factorization, synthetic division, and how to determine the sum of other roots. Suitable for advanced high school students.

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