How to Find All the Zeros of a Polynomial

Why the zeros matter

A zero of a polynomial is a value of x that makes it equal 0. On a graph, the real zeros are where the curve crosses or touches the x-axis. Factoring, solving equations, sketching graphs, and solving inequalities all start from the zeros.

Quadratics have a formula. Cubics and quartics have formulas too, but they are long enough that nobody uses them by hand. Instead you find one zero, divide it out, and repeat until a quadratic is left.

Step 1: list the possible rational zeros

The Rational Zero Theorem says that if a polynomial with whole-number coefficients has a rational zero p/q in lowest terms, then p divides the constant term and q divides the leading coefficient.

So list the factors of the constant term, list the factors of the leading coefficient, and form every fraction p/q with both signs. The real answer may not be on the list, because a polynomial can have irrational or complex zeros, but every rational zero is guaranteed to be there.

Step 2: narrow the list with Descartes' Rule of Signs

Write the terms in order of decreasing power and count how many times the sign changes from one coefficient to the next. The number of positive real zeros equals that count or is smaller than it by an even number.

Then replace x with -x, which flips the sign of every odd-power term, and count again. That gives the possible number of negative real zeros. If the rule says there is exactly one negative zero, you can stop testing negative candidates as soon as you find it.

Step 3: test candidates with synthetic division

Synthetic division divides the polynomial by (x - r) using only its coefficients. Bring the first coefficient down, multiply by r, add to the next coefficient, and repeat. The last number is the remainder.

A remainder of 0 means r is a zero, and the other numbers in the bottom row are the coefficients of the quotient, which is one degree lower. A nonzero remainder is not wasted work: by the Remainder Theorem it is the value of the polynomial at r, which tells you which side of the axis the graph is on there.

Step 4: finish with the quadratic formula

Once the quotient is a quadratic ax^2 + bx + c, solve it with x = (-b ± sqrt(b^2 - 4ac)) / (2a), or factor it if that is quicker. This step also catches irrational and complex zeros that were never on the rational list.

A full worked example

Find every zero of f(x) = 2x^3 - 3x^2 - 11x + 6.

Candidates. The constant 6 has factors 1, 2, 3, 6 and the leading coefficient 2 has factors 1, 2. The possible rational zeros are ±1, ±2, ±3, ±6, ±1/2, and ±3/2: twelve in all.

Signs. The coefficients 2, -3, -11, 6 change sign twice, so there are 2 or 0 positive zeros. For f(-x) = -2x^3 - 3x^2 + 11x + 6 the signs change once, so there is exactly one negative zero.

Testing. Try x = 1: the bottom row is 2, -1, -12, -6, so the remainder is -6 and 1 is not a zero (and f(1) = -6). Try x = 3: the bottom row is 2, 3, -2, 0. The remainder is 0, so 3 is a zero and the quotient is 2x^2 + 3x - 2.

Finishing. For 2x^2 + 3x - 2 the discriminant is 3^2 - 4(2)(-2) = 25, so x = (-3 ± 5) / 4, which gives 1/2 and -2. The three zeros are 3, 1/2, and -2: two positive and one negative, exactly as Descartes' rule allowed.

How many zeros to expect

A polynomial of degree n has exactly n zeros when you count complex zeros and count repeated zeros by their multiplicity. A cubic therefore has three zeros, but some may be complex or repeated.

When the coefficients are real, complex zeros come in conjugate pairs such as 1 + 2i and 1 - 2i. That is why every cubic with real coefficients has at least one real zero.

A zero that appears twice, as in (x - 2)^2, makes the graph touch the axis and turn around instead of crossing it.

When no candidate works

If every candidate leaves a nonzero remainder, the polynomial has no rational zeros. x^3 - 2 is an example: its only real zero is the cube root of 2, which is irrational. At that point a graph or a numerical method is the practical way to estimate the zero.

Calculator programs for this topic

These free TI-84 Plus CE Python programs carry out the steps above. Each page explains its own math, shows a screen preview, and lets you read the source before you download it.

• DuckieDai Rational Zero Finder: Lists every possible rational zero from the leading coefficient and constant term.
• DuckieDai Descartes' Rule Solver: Counts sign changes to give the possible numbers of positive and negative zeros.
• DuckieDai Synthetic Zero Tester: Shows the full synthetic division row for a candidate zero of a cubic.
• DuckieDai Cubic Zero Tester: Tests a candidate zero of a cubic and gives the quadratic left over.
• DuckieDai Cubic Zero Solver: Divides out one known zero of a cubic and solves the rest in one run.
• DuckieDai Quartic Zero Tester: Tests a candidate zero of a quartic and gives the cubic left over.
• DuckieDai Quadratic Solver: Finishes the job with the quadratic formula, including complex roots.