https://file.aiquickdraw.com/mathbot/file/1726759489077t956zbnh.jpg

Let's analyze the image and solve the polynomial problem step by step.

We are given the polynomial:

\[
p(x) = (x + 2)(x + 1)(x - 1) + x - 1
\]

The task is to determine the correct remainder when this polynomial is divided by various polynomials, which is done using the Remainder Theorem.

### Expanding \( p(x) \):

First, expand the product:

\[
(x + 2)(x + 1)(x - 1)
\]

Step 1: Expand \( (x + 1)(x - 1) \):

\[
(x + 1)(x - 1) = x^2 - 1
\]

Step 2: Multiply by \( (x + 2) \):

\[
(x + 2)(x^2 - 1) = x(x^2 - 1) + 2(x^2 - 1) = x^3 - x + 2x^2 - 2
\]

Now, add \( x - 1 \) to this result:

\[
p(x) = x^3 - x + 2x^2 - 2 + x - 1 = x^3 + 2x^2 - 3
\]

Thus, the polynomial is:

\[
p(x) = x^3 + 2x^2 - 3
\]

### Now, let's check the remainder for each option:

1. **First option**: Remainder of \( p(x) \) when divided by \( (x + 2)(x^2 - 1) \).

The polynomial \( (x + 2)(x^2 - 1) = x^3 + 2x^2 - x - 2 \). Since this matches the form of \( p(x) \), the remainder should be zero, but the option suggests it is \( 1 - x \). This option is incorrect.

2. **Second option**: Same division, but the remainder is given as \( x - 1 \). This is also incorrect since the remainder is zero.

3. **Third option**: Remainder of \( p(x) \) divided by \( x^2 - 1 \).

Since \( p(x) = x^3 + 2x^2 - 3 \) and \( x^2 - 1 \) divides into \( x^3 + 2x^2 - 3 \), the remainder is found by performing polynomial division. The remainder is \( x - 1 \), making this option correct.

4. **Fourth option**: Remainder of \( p(x) \) divided by \( x - 1 \).

Using the Remainder Theorem, substitute \( x = 1 \) into \( p(x) \):

\[
p(1) = 1^3 + 2(1)^2 - 3 = 1 + 2 - 3 = 0
\]

So, the remainder is 0, not \( x - 1 \), making this option incorrect.

### Correct answer: The third option is correct.

Do you want any further details or clarifications? Here are some related questions:

1. What is the Remainder Theorem, and how is it applied?
2. How do you perform polynomial division step by step?
3. Can polynomial factorization help in finding remainders faster?
4. What is synthetic division, and when is it useful?
5. What are some common mistakes to avoid when using the Remainder Theorem?

**Tip**: When dividing polynomials, always check for factorization shortcuts before performing full division.

Nechť p(x) = (x + 2)(x + 1)(x - 1) + x - 1. Zbytek po dělení polynomu (x + 2)(x^2 - 1) polynomem p(x) je 1 - x.

Solve a polynomial division problem using the remainder theorem. This example focuses on dividing a polynomial p(x) = (x + 2)(x^2 - 1) and determining the remainder. Explore the remainder theorem in depth with this detailed step-by-step solution. Ideal for students in Grades 10-12.

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