Math Problem Statement

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.

Solution

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

We are given the polynomial:

p(x)=(x+2)(x+1)(x1)+x1p(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)p(x):

First, expand the product:

(x+2)(x+1)(x1)(x + 2)(x + 1)(x - 1)

Step 1: Expand (x+1)(x1)(x + 1)(x - 1):

(x+1)(x1)=x21(x + 1)(x - 1) = x^2 - 1

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

(x+2)(x21)=x(x21)+2(x21)=x3x+2x22(x + 2)(x^2 - 1) = x(x^2 - 1) + 2(x^2 - 1) = x^3 - x + 2x^2 - 2

Now, add x1x - 1 to this result:

p(x)=x3x+2x22+x1=x3+2x23p(x) = x^3 - x + 2x^2 - 2 + x - 1 = x^3 + 2x^2 - 3

Thus, the polynomial is:

p(x)=x3+2x23p(x) = x^3 + 2x^2 - 3

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

  1. First option: Remainder of p(x)p(x) when divided by (x+2)(x21)(x + 2)(x^2 - 1).

The polynomial (x+2)(x21)=x3+2x2x2(x + 2)(x^2 - 1) = x^3 + 2x^2 - x - 2. Since this matches the form of p(x)p(x), the remainder should be zero, but the option suggests it is 1x1 - x. This option is incorrect.

  1. Second option: Same division, but the remainder is given as x1x - 1. This is also incorrect since the remainder is zero.

  2. Third option: Remainder of p(x)p(x) divided by x21x^2 - 1.

Since p(x)=x3+2x23p(x) = x^3 + 2x^2 - 3 and x21x^2 - 1 divides into x3+2x23x^3 + 2x^2 - 3, the remainder is found by performing polynomial division. The remainder is x1x - 1, making this option correct.

  1. Fourth option: Remainder of p(x)p(x) divided by x1x - 1.

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

p(1)=13+2(1)23=1+23=0p(1) = 1^3 + 2(1)^2 - 3 = 1 + 2 - 3 = 0

So, the remainder is 0, not x1x - 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.

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Math Problem Analysis

Mathematical Concepts

Polynomial Division
Remainder Theorem
Algebra

Formulas

Remainder Theorem: p(x) = (x + 2)(x^2 - 1)

Theorems

Polynomial Remainder Theorem

Suitable Grade Level

Grades 10-12