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 September 10th, 2019, 11:47 AM #1 Newbie   Joined: Jan 2017 From: London Posts: 18 Thanks: 0 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.
 September 10th, 2019, 11:58 AM #2 Senior Member   Joined: Jun 2019 From: USA Posts: 376 Thanks: 202 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 $\displaystyle a_0, a_1, ...$).
 September 10th, 2019, 12:07 PM #3 Newbie   Joined: Jan 2017 From: London Posts: 18 Thanks: 0 Sorry my bad. It's $\displaystyle S=(x-1)^4+4(x-1)^3+6(x-1)^2+4(x-1)+1$
 September 10th, 2019, 12:25 PM #4 Senior Member   Joined: Jun 2019 From: USA Posts: 376 Thanks: 202 Yes, just expand the brackets and combine like terms. Thanks from mathsonlooker
 September 10th, 2019, 12:28 PM #5 Senior Member     Joined: Sep 2015 From: USA Posts: 2,636 Thanks: 1472 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 $S = ((x-1) + 1)^4 = x^4$ Thanks from mathsonlooker Last edited by romsek; September 10th, 2019 at 12:36 PM.
September 10th, 2019, 12:30 PM   #6
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Quote:
 Originally Posted by romsek 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.

September 10th, 2019, 12:35 PM   #7
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Quote:
 Originally Posted by DarnItJimImAnEngineer Shhh! You're supposed to let them discover that on their own.

September 10th, 2019, 12:44 PM   #8
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Quote:
 Originally Posted by DarnItJimImAnEngineer Shhh! You're supposed to let them discover that on their own.
Evil

September 10th, 2019, 12:47 PM   #9
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Quote:
 Originally Posted by romsek 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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