Let P₂ be the vector space of polynomials of degree at most 2. Determine the coordinates of the polynomial p(x) = x+2²-2 relative to the basis B = {1,2-2, x² — x}.
Let P₂ be the vector space of polynomials of degree at most 2. Determine the coordinates of the polynomial p(x) = x+2²-2 relative to the basis B = {1,2-2, x² — x}.
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter1: Fundamental Concepts Of Algebra
Section1.3: Algebraic Expressions
Problem 32E
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![**Problem Statement:**
Let \( P_2 \) be the vector space of polynomials of degree at most 2. Determine the coordinates of the polynomial \( p(x) = x + x^2 - 2 \) relative to the basis \( B = \{1, x - 2, x^2 - x\} \).
**Explanation:**
We need to express the polynomial \( p(x) = x + x^2 - 2 \) in terms of the basis \( B \). This requires us to find the coefficients \( c_1, c_2, \) and \( c_3 \) such that:
\[ p(x) = c_1 \cdot 1 + c_2 \cdot (x - 2) + c_3 \cdot (x^2 - x) \]
where \( 1, (x - 2), \) and \( (x^2 - x) \) form the basis \( B \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdf4abcde-0790-4b6d-b8ff-52c0ea5b10aa%2Fedf70654-3f8a-4d3c-a3aa-a4f2d347d6e9%2F9z9bv1_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Let \( P_2 \) be the vector space of polynomials of degree at most 2. Determine the coordinates of the polynomial \( p(x) = x + x^2 - 2 \) relative to the basis \( B = \{1, x - 2, x^2 - x\} \).
**Explanation:**
We need to express the polynomial \( p(x) = x + x^2 - 2 \) in terms of the basis \( B \). This requires us to find the coefficients \( c_1, c_2, \) and \( c_3 \) such that:
\[ p(x) = c_1 \cdot 1 + c_2 \cdot (x - 2) + c_3 \cdot (x^2 - x) \]
where \( 1, (x - 2), \) and \( (x^2 - x) \) form the basis \( B \).
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