---
title: "Express x³−5x²+x+2 in Legendre polynomials"
subject: "Mathematics"
chapter: 4
key: "semester_2/mathematics:express-x3-5x2-x-2"
source: https://www.bctengineer.com/q/semester_2/mathematics/express-x3-5x2-x-2
site: BCT Engineering
---

# Express x³−5x²+x+2 in Legendre polynomials

Mathematics · Chapter 4: Solution of Differential Equations in Series and Special Functions

- Asked in: 2073 Bhadra, 5 marks; 2072 Magh, 5 marks; 2070 Bhadra, 5 marks (asked 3 times)
- Video: [Express f(x) interms of Legendre's polynomials - Example Problem 3 | Engineering Mathematics](https://www.youtube.com/watch?v=ocqWh8n98v4) — Mathematics Tutor
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_2/mathematics/express-x3-5x2-x-2
- Whole chapter: https://www.bctengineer.com/md/semester_2/mathematics/chapter-4

## Question

Express f(x) = x³ − 5x² + x + 2 in terms of Legendre's polynomial.

## Short answer

Express powers using P_0=1, P_1=x, P_2=(3x²-1)/2, P_3=(5x³-3x)/2. Substitution yields f(x) = (2/5)P_3(x) - (10/3)P_2(x) + (8/5)P_1(x) + (1/3)P_0(x).

## Answer

Use the Legendre polynomials P₀ = 1, P₁ = x, P₂ = (3x² − 1)/2, P₃ = (5x³ − 3x)/2.

#### Express powers of x in Legendre polynomials
```text
x³ = (2P₃ + 3P₁)/5      because P₃ = (5x³ − 3x)/2  →  x³ = (2P₃ + 3x)/5
x² = (2P₂ + P₀)/3       because P₂ = (3x² − 1)/2   →  x² = (2P₂ + 1)/3
x  = P₁
1  = P₀
```

#### Substitute
f(x) = x³ − 5x² + x + 2

```text
= (2/5)P₃ + (3/5)P₁ − (5/3)(2P₂ + P₀) + P₁ + 2P₀
= (2/5)P₃ − (10/3)P₂ + (3/5 + 1)P₁ + (2 − 5/3)P₀
= (2/5)P₃ − (10/3)P₂ + (8/5)P₁ + (1/3)P₀
```

**Answer: x³ − 5x² + x + 2 = (2/5)P₃(x) − (10/3)P₂(x) + (8/5)P₁(x) + (1/3)P₀(x).**

Check at x = 1 (all Pₙ(1) = 1): 2/5 − 10/3 + 8/5 + 1/3 = 2 − 3 = −1, and f(1) = 1 − 5 + 1 + 2 = −1 ✓.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. x^3 in terms of Legendre polynomials is:
   A. (2P3 + 3P1)/5
   B. (2P3 - 3P1)/5
   C. (5P3 + 3P1)/2
   D. P3 + P1
   Answer: A. From P3 = (5x^3-3x)/2. (Key point: x^3)

2. x^2 in Legendre polynomials is:
   A. (2P2 - P0)/3
   B. (2P2 + P0)/3
   C. (P2 + P0)/2
   D. (3P2 + P0)/2
   Answer: B. From P2 = (3x^2-1)/2. (Key point: x^2)

3. The coefficient of P2 in the result is:
   A. -5/3
   B. 10/3
   C. -10/3
   D. -5
   Answer: C. -5 * (2/3). (Key point: Coefficient of P2)

4. The full expansion is:
   A. (2/5)P3 - (10/3)P2 + (8/5)P1 + (1/3)P0
   B. (2/5)P3 - (10/3)P2 + (8/5)P1 - (1/3)P0
   C. (2/5)P3 + (10/3)P2 + (8/5)P1 + (1/3)P0
   D. (2/5)P3 - (10/3)P2 + (3/5)P1 + (1/3)P0
   Answer: A. P1: 3/5 + 1 = 8/5. P0: 2 - 5/3 = 1/3. (Key point: Final)

## Related in Mathematics

- [Express x³−5x²+6x+1 in Legendre polynomials](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Aexpress-x3-5x2-6x-1) — Mathematics, chapter 4, asked 3×
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