Polynomials

A

Albi

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May 9, 2020
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Determine a, b and c so that

p(x) = 2x^3 - 9x^2 + 13x - 6

equals g(x) = (x-2)(ax^2 + bx + c)
I have found the parameters they are a = 2, b= -5 , c = 3 just by casually trying
But is there any way to solve this problem algebraically.
 
Albi said:
Determine a, b and c so that

p(x) = 2x^3 - 9x^2 + 13x - 6

equals g(x) = (x-2)(ax^2 + bx + c)
I have found the parameters they are a = 2, b= -5 , c = 3 just by casually trying
But is there any way to solve this problem algebraically.
Click to expand...
Yes ....

expand g(x) = (x-2)(ax^2 + bx + c) and write it as a polynomial.

Now compare and equate coefficients of equivalent powers of x in p(x and g(x) - and derive your solutions
 
HallsofIvy said:
"Casually trying" is a perfectly good way to solve problems!
Click to expand...
I very slightly disagree with Halls. Not about mathematics, but about his exact choice of words.

A different helper here frequently and wisely says “correct answers do not care how they are found.”

Some types of problem have no known solution other than trying. There actually is a field of study in mathematics on using “trying” to get answers; it is given the fancy name of “numerical methods.”

Computers make experimentation far less burdensome than it once was. Furthermore, computers can easily make some mathematical ideas (such as convergence on a limit) intuitive.

In short, trying, also known as experimentation, is a valuable tool in mathematics.

So in what respects do I disagree with Halls?

I would prefer saying:

“Casually trying is a perfectly acceptable way to solve problems.”

I would not say “perfectly good” because “casually” is likely to result in a ton of fruitless work.

Systematic trying is much more likely to reach a correct answer sooner and with far less work. We should teach methods of systematic trying long before a course on numerical methods.
 
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