What can you tell about c from knowledge of f(0) ?lolo94 said:Homework Statement
Let f(x) = ax^2 + bx + c be a quadratic polynomial. Either prove or disprove the following statement: If f(0) and f(1) are even integers then f(n) is an integer for every natural number n.
Homework Equations
The Attempt at a Solution
I tried different approaches such as analyzing the constants, f(n)-f(0)-f(1).
How do you approach these problems in general?
c=even integerSammyS said:What can you tell about c from knowledge of f(0) ?
The purpose of proving even integer coefficients in quadratic polynomials is to ensure that the polynomial is factorable and can be solved using the quadratic formula or factoring methods. It also helps to verify the accuracy of the polynomial and its solutions.
To prove that a polynomial has even integer coefficients, you can use the Rational Root Theorem. This theorem states that if a polynomial has integer coefficients, then any rational root must be in the form of p/q, where p is a factor of the constant term and q is a factor of the leading coefficient. By testing these possible rational roots, you can determine if the polynomial has even integer coefficients.
Yes, a polynomial can have even integer coefficients but not be factorable. This is because even if a polynomial has integer coefficients, it may not have any rational roots. In this case, the polynomial would need to be solved using the quadratic formula or other methods instead of factoring.
The even integer coefficients in a quadratic polynomial affect the graph by determining the direction and shape of the parabola. If the leading coefficient (a) is positive, the parabola will open upward and if it is negative, the parabola will open downward. The value of the constant term (c) also affects the y-intercept of the parabola.
Solving for even integer coefficients in quadratic polynomials is important because it helps to ensure that the polynomial can be accurately solved and analyzed. It also allows for the use of various methods, such as factoring, to solve the polynomial. Additionally, knowing the even integer coefficients can provide valuable information about the graph and behavior of the polynomial.