Proving Even Integer Coefficients in Quadratic Polynomials - Homework Question

In summary, the conversation discusses a quadratic polynomial and the statement that if f(0) and f(1) are even integers, then f(n) is also an integer for every natural number n. Different approaches were tried, including analyzing the constants and using the function to construct a parabola through three points. It is suggested to use f(n1) as the third point to test the statement.
  • #1
lolo94
17
0

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?
 
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  • #2
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?
What can you tell about c from knowledge of f(0) ?
 
  • #3
SammyS said:
What can you tell about c from knowledge of f(0) ?
c=even integer
 
  • #4
We can construct a unique parabola using 3 points. Consider the function f$$x → ax^2 + bx + c$$

We know:
f(0) = a1
f(1) = a2

a1 and a2 are even integers. You can use f(n1) for the third point. Then you have a parabola through these 3 points. Try this.
 
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Likes SammyS

1. What is the purpose of proving even integer coefficients in quadratic polynomials?

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.

2. How do you prove that a polynomial has even integer coefficients?

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.

3. Can a polynomial have even integer coefficients but not be factorable?

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.

4. How do even integer coefficients affect the graph of a quadratic polynomial?

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.

5. Why is it important to solve for even integer coefficients in quadratic polynomials?

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.

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