'Equating the coefficients' question

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Homework Help Overview

The discussion revolves around expanding a cubic expression derived from the product of two polynomials, specifically the expression (x-3)(px^2+qx+r). Participants are examining the process of expansion and the subsequent collection of like terms to arrive at a specific form of the cubic polynomial.

Discussion Character

  • Exploratory, Mathematical reasoning, Problem interpretation

Approaches and Questions Raised

  • Participants discuss the method of expanding the brackets and collecting like terms. Questions arise regarding the factorization of the resulting polynomial and how specific terms are derived from the expansion process.

Discussion Status

Some participants are clarifying their understanding of the expansion and factorization steps, while others are reflecting on their observations regarding the structure of the cubic polynomial. There is an ongoing exploration of how to manipulate the terms correctly without reaching a definitive conclusion.

Contextual Notes

Participants are working within the constraints of a textbook example and are attempting to reconcile their understanding with the provided solution, which may not be fully clear to them.

Parsifal1
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Homework Statement


Expand the brackets to get a cubic containing the unknowns. This is an example in the textbook but I don't see how they've expanded the brackets to get their answer:

(x-3)(px^2+qx+r)=px^3+(q-3p)x^2+(r-3q)x-3r

Homework Equations


(x-3)(px^2+qx+r)

The Attempt at a Solution



I would have done it how you expand long brackets:

(x-3)(px^2)+(x-3)(qx)+(x-3)(r)=px^3-3px^2...
 
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Yes, and the next step is to collect all the terms involving x^2 together, and all the terms involving x together...
 
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pasmith said:
Yes, and the next step is to collect all the terms involving x^2 together, and all the terms involving x together...

What I don't get is how they factorized what you get from expanding: px^3+qx^2+rx-3px^2-3qx-3r. How do you get px^3+(q-3p)x^2+(r-3q)x-3r from that?
 
Ah, I see if you do (q-3p)x^2 and multiply it out and the same with the other factorized terms, you get the terms you get when you'd first expand it out. I hadn't noticed that px^3+(q-3p)x^2... etc. was a cubic. I need to work on noticing things, I've found. :rolleyes:
 
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