Real Zeros of Polynomials

Real zeros of a polynomial are the input values for which the polynomial equals zero. Rational zeros are a subset of real zeros that can be expressed as a ratio of two integers.

The Rational Zeros Theorem states that in a polynomial with integer coefficients, any rational zero must be a fraction p over q, where p divides the constant term and q divides the leading coefficient.

The Rational Zeros Theorem identifies possible rational zeros by dividing the factors of the constant term, which are the numbers that evenly divide the constant, by the factors of the coefficient of the variable with the highest power.

These possible values are tested using synthetic division, where the polynomial’s coefficients are divided by each candidate real zero. A nonzero remainder means the value is not a zero of the polynomial. A zero remainder confirms an actual zero.

In packaging design, the volume of a box can be modeled by a polynomial in terms of cut size. Setting this polynomial equal to a target volume and rearranging gives a new equation equal to zero. The real zeros represent the cut sizes that achieve the desired capacity without wasting material.