Forming A Quadratic Polynomial From Zeroes

To construct a quadratic polynomial when given the sum and product of its zeroes, we rely on the standard relationships. The sum of the zeroes is equal to $-b/a$ and their product is equal to $c/a$. In this exercise, the given sum is $1/4$ and the product is $-1$. To avoid fractions, we can smartly choose the denominator as our leading coefficient by letting $a = 4$. Setting $-b/4 = 1/4$ simplifies to $b =$ . Similarly, setting $c/4 = -1$ gives us $c =$ . Substituting $a$, $b$, and $c$ into the general form $ax^2 + bx + c$ yields the final polynomial. Therefore, one valid quadratic polynomial that fits these conditions is $4x^2 - x +$ .