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anemone
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Find a polynomial of degree 3 with real coefficients such that each of its roots is equal to the square of one root of the polynomial $P(x)=x^3+9x^2+9x+9$.
Does the above mean that if a,b,c are the 3 roots of the above polynomial then the the roots of the new polynomial must be $a^2 ,b^2 , c^2$ ??anemone said:Find a polynomial of degree 3 with real coefficients such that each of its roots is equal to the square of one root of the polynomial $P(x)=x^3+9x^2+9x+9$.
kaliprasad said:Because the square of the roots shall be roots of the required polynomial so we must have $\sqrt x$ as roots of P(X)
So $(\sqrt x)^3 + 9 x + 9 (\sqrt x) + 9 = 0$
Or $(\sqrt x)(x + 9) = - 9(x+1)$
Or squaring $x(x+9)^2 = 81(x+1)^2$
Or $x(x^2 + 18 x + 81) = 81 x^2 + 162x + 81$
Or $x^3 - 63x^2 - 81 x - 81 = 0$
This is the required equation
solakis said:[sp]kaliprasand must we not have also $ -\sqrt x$ .Anyway i did this problem in a different way and i got the same answer [/sp]
A cubic function is a type of polynomial function with a degree of 3. It is written in the form f(x) = ax^3 + bx^2 + cx + d, where a, b, c, and d are constants and x is the independent variable.
To find the cubic function of a set of data points, you can use a process called regression analysis. This involves finding the best-fitting curve that passes through all of the data points. You can use software or a graphing calculator to perform this analysis.
A cubic function is a type of polynomial function, specifically one with a degree of 3. A polynomial of degree 3 can have a variety of forms, including quadratic, cubic, or quartic functions. So, while all cubic functions are polynomials of degree 3, not all polynomials of degree 3 are cubic functions.
A cubic function can have up to 3 solutions, depending on the values of its coefficients. However, it is possible for a cubic function to have fewer solutions or no solutions at all. This can be determined by analyzing the graph of the function or by solving the function algebraically.
Yes, a cubic function can have a negative leading coefficient. This means that the coefficient of the highest degree term (ax^3) can be negative. This can result in a downward-facing graph, as opposed to the typical upward-facing graph of a cubic function with a positive leading coefficient.