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mangren
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Homework Statement
A cubic polynomial gives remainders (5x + 4) and (12x -1) when divided by x^2 - x + 2 and x^2 + x - 1 respectively. Find the polynomial.
I presume that you seek help with this problem, rather than merely intending to duplicate your textbook?mangren said:Homework Statement
A cubic polynomial gives remainders (5x + 4) and (12x -1) when divided by x^2 - x + 2 and x^2 + x - 1 respectively. Find the polynomial.
No, that's wrong. P(x)/(x^2- x+ 2)= Q(x)+ (5x+4)/(x^2- x+ 2) andmangren said:Sorry, this is the first time I posted here. I'm lost in this problem. I tried starting solving it by using:
P(x)/(x^2 - x + 2) = Q(x) + (5x + 4) and
P(x)/(x^2 + x - 1) = R(x) + (12x - 1)
I see four dimensions of equations and four unknowns; there shouldn't be a problem. Don't forget that, for two polynomials to be equal, their coefficients must be equal!mangren said:...
x(b + c) -a(x + 2) - 2b + d which is equal to (5x + 4),
...
x(2a + c) - bx + d - a which is also equal to (12x - 1), then i tried to solve simultaneously, but there are just too many unknowns... don't know what to do next...
HallsofIvy said:It might help to write it as P(x)= Q(x)(x^2- x+ 2)+ 5x+4 and P(x)= R(x)(x^2+ x-1)+ 12x-1.
You also know that P is cubic so Q and R must be linear. P(0)= 2Q(0)+ 4= -R(0)- 1 and P(1)= 2Q(1)+ 9= R(x)+ 11. Two points should be enough to determin a linear equation.
To find a cubic polynomial with remainders, you need to have at least four data points (x and y values) and then use the method of polynomial interpolation to determine the coefficients of the polynomial.
The method of polynomial interpolation is a mathematical technique used to find a polynomial that passes through a given set of data points. It involves solving a system of equations using the data points to determine the coefficients of the polynomial.
No, the four data points should be distinct and not lie on the same line. If the data points are collinear, it will not be possible to find a unique cubic polynomial that passes through all the points.
The degree of the cubic polynomial found using the method of polynomial interpolation will always be three, as it is a cubic polynomial. This means that the highest power of the variable in the polynomial will be three.
Yes, this method may not work if the data points are too far apart or if there is a lot of noise in the data. In such cases, the polynomial found may not accurately represent the data points. It is important to have a good understanding of the data and the limitations of this method before using it.