https://file.aiquickdraw.com/mathbot/file/17296174739020w0bjeb3.jpg

The given problem asks to find the polynomial \( P(x+1) \) based on the polynomial \( P(x^3) \), which is:

\[
P(x^3) = (a+2)x^7 + (b-1)x^6 + (b+3)x^5 + (a+b)x^3 + b
\]

The steps to solve this problem involve substituting \( x+1 \) into the polynomial in place of \( x \). Let's work through the substitution.

### Step 1: Substitute \( x + 1 \) into \( P(x^3) \)

The structure of \( P(x^3) \) will remain the same in terms of powers of \( x \), but now we replace \( x \) with \( (x+1) \).

The terms become:
\[
P(x+1) = (a+2)(x+1)^7 + (b-1)(x+1)^6 + (b+3)(x+1)^5 + (a+b)(x+1)^3 + b
\]

### Step 2: Expand each binomial expression

We will expand each term one by one using binomial expansion.

#### For \( (x+1)^7 \):
\[
(x+1)^7 = x^7 + 7x^6 + 21x^5 + 35x^4 + 35x^3 + 21x^2 + 7x + 1
\]
So:
\[
(a+2)(x+1)^7 = (a+2)(x^7 + 7x^6 + 21x^5 + 35x^4 + 35x^3 + 21x^2 + 7x + 1)
\]

#### For \( (x+1)^6 \):
\[
(x+1)^6 = x^6 + 6x^5 + 15x^4 + 20x^3 + 15x^2 + 6x + 1
\]
So:
\[
(b-1)(x+1)^6 = (b-1)(x^6 + 6x^5 + 15x^4 + 20x^3 + 15x^2 + 6x + 1)
\]

#### For \( (x+1)^5 \):
\[
(x+1)^5 = x^5 + 5x^4 + 10x^3 + 10x^2 + 5x + 1
\]
So:
\[
(b+3)(x+1)^5 = (b+3)(x^5 + 5x^4 + 10x^3 + 10x^2 + 5x + 1)
\]

#### For \( (x+1)^3 \):
\[
(x+1)^3 = x^3 + 3x^2 + 3x + 1
\]
So:
\[
(a+b)(x+1)^3 = (a+b)(x^3 + 3x^2 + 3x + 1)
\]

### Step 3: Collect all the expanded terms

After expanding all the terms, we combine them to form the full expression of \( P(x+1) \).

### Would you like me to continue with the full expansion and simplification of the terms or explain any specific part of the expansion process?

P(x^3) = (a+2)x^7 + (b-1)x^6 + (b+3)x^5 + (a+b)x^3 + b olduğuna göre, P(x+1) polinomunu bulunuz.

In this problem, we are given a polynomial P(x^3) = (a+2)x^7 + (b-1)x^6 + (b+3)x^5 + (a+b)x^3 + b and asked to find P(x+1). The solution involves substituting x+1 into the original polynomial and using binomial expansion to simplify the terms. This problem is suitable for high school students learning about polynomial manipulation and binomial expansion.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation