20160728, 02:12  #1 
veganjoy
"Joey"
Nov 2015
Middle of Nowhere,AR
3·5·29 Posts 
Polynomial Problem
I'm in an Algebra II class, and the current topic is polynomials. I was working on a problem, but I have no idea where to begin.
Suppose \(f\) is a polynomial such that \(f(0)=47, f(1)=32, f(2)=13,\) and \(f(3)=16.\) What is the sum of the coefficients of \(f\)? Clearly, to find the sum of the coefficients, I need to find the function \(f\). I'm not sure how to find \(f\) given a set of points. One potential option I saw on the Internet was polynomial interpolation, but the Wikipedia article was very complicated. Is there a simpler explanation of this technique, if it is the right approach? If not, what is the right approach to this problem? Thanks! 
20160728, 04:14  #2 
Apr 2012
364_{10} Posts 
f(x)=a*x^4+b*x^3+c*x^2+d*x+47, since there are 4 roots assuming a single variable.
Substitute the root given for x in four separate equations then solve for the coefficients. Check all roots via substitution once the function is obtained then sum your coefficients. This is how I would start. Last fiddled with by jwaltos on 20160728 at 04:21 
20160728, 04:21  #3 
Jun 2003
2×3^{2}×269 Posts 

20160728, 04:24  #4 
"Curtis"
Feb 2005
Riverside, CA
17×271 Posts 
Why are there four roots? I see evidence for two. With 4 ordered pairs given, create a polynomial with four unknown coefficients, including the constant term. Since f(0) = 47, 47 is the constant term. That leaves three unknown coeffs in a cubic (degree 3).

20160728, 04:24  #5 
Apr 2012
2^{2}×7×13 Posts 
Probably, Jack Daniels, Jim Beam and Johnny Walker were all giving advice.

20160728, 04:43  #6 
"Antonio Key"
Sep 2011
UK
3^{2}×59 Posts 
Surely f(1) = 32 = the sum of the coefficients of f(x) for a polynomial of any degree, as all x^{n} = 1 so we are reduced to a_{n}*1 + a_{n1}*1 + a_{n2}*1 + ........ + a_{0} = 32

20160728, 05:27  #7 
Jun 2003
2·3^{2}·269 Posts 

20160728, 13:44  #8 
veganjoy
"Joey"
Nov 2015
Middle of Nowhere,AR
3·5·29 Posts 
I think I get the idea. So the polynomial must be of the form \(ax^3+bx^2+cx+d,\) since there are 4 given points, and one of them is the \(y\) intercept.
So if \(x=1,\) all of the \(x\) values stay the same since \(1\) to any power is \(1\). However, if \(f(1)=32,\) then that is \(a+b+c+47,\) so I would need to subtract \(47\) from \(a+b+c\) to get \(7\) as the answer. Either that or I overthought this problem. Last fiddled with by jvang on 20160728 at 13:47 
20160728, 17:08  #9 
(loop (#_fork))
Feb 2006
Cambridge, England
7×911 Posts 
Whatever the polynomial, f(1) is the sum of its coefficients. You weren't given the degree of the polynomial, because you don't need it.
(because if f(x) = sum a_i x^i, then f(1) = sum a_i because all the powers of 1 are 1) Equally f(1) is the alternating sum of its coefficients, so f(1)+f(1)/2 is the sum of the evennumbered coefficients. This turns into a root into Fourier analysis: think about $f(\omega)$ where $\omega$ is a root of unity. Last fiddled with by fivemack on 20160728 at 17:09 
20160729, 03:35  #10 
Romulan Interpreter
Jun 2011
Thailand
3×11×277 Posts 
@OP: Are you sure the problem asks for the coefficient's sum? As the problem is given, it is either tricky (because you only need to know f(1) and nothing else, no information about the degree is necessary, and the other 3 values given are futile) or either it wanted to ask for the sum of the roots. In this case, you need all 4 values, make s system with 4 equations and 4 variables, check if they are not linear (i.e. the system may not be independent, in such case a lowerdegree poly will fit) and/or use Vieta's formulas to get the sum.
Last fiddled with by LaurV on 20160729 at 03:36 Reason: link 
20160729, 14:46  #11 
Aug 2006
1011101001001_{2} Posts 

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