Find all possible and actual rational zeros of a polynomial using the Rational Zeros Theorem. Step-by-step solutions.
Want to solve ANY math problem? Just take a screenshot.
Math.Photos handles algebra, calculus, geometry, statistics, and more — all from a photo.
If a polynomial has integer coefficients, any rational zero p/q must have p as a factor of the constant term and q as a factor of the leading coefficient.
List all factors of the constant term (p) and all factors of the leading coefficient (q). Form all possible fractions p/q with both positive and negative signs.
Substitute each candidate into the polynomial. If f(p/q) = 0, then p/q is a rational zero. Use synthetic division for efficiency.
Once you find a rational zero, factor it out using synthetic division to reduce the polynomial degree. Repeat until all zeros are found.
Math.Photos solves any math problem from a screenshot — algebra, calculus, geometry, statistics, and more. With step-by-step explanations.
InstallUse the Rational Zeros Theorem: list all factors p of the constant term and all factors q of the leading coefficient. The possible rational zeros are all values of the form p/q (with both + and - signs). Then test each candidate by plugging it into the polynomial.
The Rational Zeros Theorem states that if a polynomial f(x) = a_n*x^n + ... + a_1*x + a_0 has integer coefficients, then every rational zero has the form p/q, where p divides a_0 (constant term) and q divides a_n (leading coefficient).
Yes. For example, x^2 + 1 = 0 has no real zeros at all (only complex: x = i and x = -i). And x^2 - 2 = 0 has zeros x = +/-sqrt(2), which are irrational. The Rational Zeros Theorem only finds rational zeros.
Once you find that x = r is a zero, divide the polynomial by (x - r) using synthetic division. Write r on the left, the coefficients across the top, bring down the first, multiply-and-add across. The result is a polynomial of one lower degree. Repeat to find remaining zeros.