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anemone
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Let $p, q, r, s \in \mathbb{N}$ such that $p \ge q \ge r \ge s$. Show that the function $f(x)=x^4-px^3-qx^2-rx-s$ has no integer root.
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[sp]Suppose that $n$ is an integer such that $f(n) = 0$. Then $n\ne0$, because $s\ne0$.anemone said:Let $p, q, r, s \in \mathbb{N}$ such that $p \ge q \ge r \ge s$. Show that the function $f(x)=x^4-px^3-qx^2-rx-s$ has no integer root.
Integer roots refer to solutions of a function that are whole numbers, meaning they are positive or negative numbers with no decimal or fractional parts.
Yes, a function can have both integer and non-integer roots. For example, the function f(x) = x^2 - 4 has integer roots of -2 and 2, but also has non-integer roots of ±2i.
One way to prove that a function has no integer roots is to use the Rational Root Theorem. This theorem states that if a polynomial function has integer roots, they must be factors of the constant term divided by the leading coefficient. By checking all possible factors using synthetic division, you can determine if any of them are roots of the function. If none of the possible factors are roots, then the function has no integer roots.
Yes, there are other methods to prove that a function has no integer roots. For example, you can use the Intermediate Value Theorem to show that the function does not cross the x-axis at any integer points, or you can use a graphing calculator to visually determine if the function has any integer roots.
Proving that a function has no integer roots can help determine the behavior and properties of the function. It can also be useful in solving equations or inequalities involving the function, as it narrows down the possible solution set. Additionally, it can provide insight into the relationship between the function's coefficients and its roots.