Find the remainder when the polynomial x17 + x15 + x° + x + 1 is divided by x3 - x.

To find the remainder when the polynomial \( f(x) = x^{17} + x^{15} + x^0 + x + 1 \) is divided by \( g(x) = x^3 - x \), we can use polynomial division. However, an efficient method is to leverage the fact that \( g(x) = x^3 - x = x(x^2 - 1) \), which has roots \( x = 0, 1, -1 \).

Thus, the remainder can be expressed as a polynomial of degree less than 3, so we can write the remainder as \( r(x) = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants.

### Step 1: Evaluate \( f(x) \) at the roots of \( g(x) \) to find \( a \), \( b \), and \( c \).

#### Root 1: \( x = 0 \)
Substitute \( x = 0 \) into \( f(x) \):
\[
f(0) = 0^{17} + 0^{15} + 0^0 + 0 + 1 = 1
\]
Thus, \( r(0) = 1 \), so \( c = 1 \).

#### Root 2: \( x = 1 \)
Substitute \( x = 1 \) into \( f(x) \):
\[
f(1) = 1^{17} + 1^{15} + 1^0 + 1 + 1 = 1 + 1 + 1 + 1 + 1 = 5
\]
Thus, \( r(1) = 5 \), so \( a + b + c = 5 \).

#### Root 3: \( x = -1 \)
Substitute \( x = -1 \) into \( f(x) \):
\[
f(-1) = (-1)^{17} + (-1)^{15} + (-1)^0 + (-1) + 1 = -1 + (-1) + 1 - 1 + 1 = -1
\]
Thus, \( r(-1) = -1 \), so \( a - b + c = -1 \).

### Step 2: Solve the system of equations.
We now have the following system of equations:
1. \( c = 1 \)
2. \( a + b + 1 = 5 \)  (from \( r(1) = 5 \))
3. \( a - b + 1 = -1 \)  (from \( r(-1) = -1 \))

Substitute \( c = 1 \) into the second and third equations:
- \( a + b + 1 = 5 \) becomes \( a + b = 4 \)
- \( a - b + 1 = -1 \) becomes \( a - b = -2 \)

Now solve the system:
- Adding the two equations: \( (a + b) + (a - b) = 4 + (-2) \), which gives \( 2a = 2 \), so \( a = 1 \).
- Substituting \( a = 1 \) into \( a + b = 4 \), we get \( 1 + b = 4 \), so \( b = 3 \).

### Step 3: Write the remainder.
The remainder is:
\[
r(x) = ax^2 + bx + c = 1x^2 + 3x + 1 = x^2 + 3x + 1
\]

Thus, the remainder when \( f(x) = x^{17} + x^{15} + x^0 + x + 1 \) is divided by \( g(x) = x^3 - x \) is \( \boxed{x^2 + 3x + 1} \).

Would you like to see more details or have any questions?

Here are 5 related questions:
1. What is the degree of a remainder when dividing any polynomial by a cubic polynomial?
2. How can you find the remainder using synthetic division for polynomials?
3. What are the properties of polynomials that allow for efficient division by simple polynomials?
4. Can the remainder theorem be applied in this case? Why or why not?
5. How do you find the roots of a cubic polynomial like \(x^3 - x\)?

**Tip:** When dividing polynomials, finding roots of the divisor can often simplify the calculation of the remainder.

Find the remainder when the polynomial x^17 + x^15 + x^0 + x + 1 is divided by x^3 - x.

Learn how to find the remainder when the polynomial x^17 + x^15 + x + 1 is divided by x^3 - x using polynomial division and the remainder theorem. This detailed step-by-step solution covers key concepts like roots of polynomials and solving systems of equations, suitable for Grades 10-12.

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