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September 10th, 2019, 11:47 AM   #1
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What is this question asking for? Polynomial

Question:

Find S as a polynomial in x where

$\displaystyle
S=(x-1)^4+4(x-1)^3+6(x-1)^2+4(x-1)+1
$


Understanding what is being asked here would be a useful first step

Do I need to expand the brackets or what?

I understand a polynomial to be an expression involving sums of powers of a variable e.g. x and the powers have to be non negative to qualify as a polynomial

Last edited by mathsonlooker; September 10th, 2019 at 12:06 PM.
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September 10th, 2019, 11:58 AM   #2
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I think there's something missing in the wording. "Where," makes sense when followed by an equation, inequality, or condition, not an expression.

But yes, my guess is they just want you to expand that and write the expression as
$\displaystyle S=a_4 x^4 + a_3 x^3 +a_2 x^2 + a_1 x + a_0$ (find $\displaystyle a_0, a_1, ...$).
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September 10th, 2019, 12:07 PM   #3
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Sorry my bad. It's

$\displaystyle
S=(x-1)^4+4(x-1)^3+6(x-1)^2+4(x-1)+1
$
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September 10th, 2019, 12:25 PM   #4
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Yes, just expand the brackets and combine like terms.
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September 10th, 2019, 12:28 PM   #5
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Basically they want to you expand the whole thing out and then regroup all the terms in powers of $x$

That being said. The structure of the coefficients (they are binomial coefficients) suggests
some sort of mathematical sleight of hand to make this problem less brain numbing.

$S = \sum \limits_{k=0}^4 \dbinom{4}{k}(x-1)^k \cdot 1^{4-k}$

Using the binomial theorem this is

$S = ((x-1) + 1)^4 = x^4$
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Last edited by romsek; September 10th, 2019 at 12:36 PM.
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September 10th, 2019, 12:30 PM   #6
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Quote:
Originally Posted by romsek View Post
Using the binomial theorem this is

$S = ((x-1) + 1)^4 = x^4$
Shhh! You're supposed to let them discover that on their own.
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September 10th, 2019, 12:35 PM   #7
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Quote:
Originally Posted by DarnItJimImAnEngineer View Post
Shhh! You're supposed to let them discover that on their own.
sadist
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September 10th, 2019, 12:44 PM   #8
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Quote:
Originally Posted by DarnItJimImAnEngineer View Post
Shhh! You're supposed to let them discover that on their own.
Evil
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September 10th, 2019, 12:47 PM   #9
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Quote:
Originally Posted by romsek View Post
Basically they want to you expand the whole thing out and then regroup all the terms in powers of $x$

That being said. The structure of the coefficients (they are binomial coefficients) suggests
some sort of mathematical sleight of hand to make this problem less brain numbing.

$S = \sum \limits_{k=0}^4 \dbinom{4}{k}(x-1)^k \cdot 1^{4-k}$

Using the binomial theorem this is

$S = ((x-1) + 1)^4 = x^4$
Wow!
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