How to Find Zeros of Polynomials: Steps and Examples

How to find polynomial zeros step by step

1. Set the output equal to zero. A zero is an input \(c\) satisfying \(f(c)=0\). The corresponding real-coordinate intercept is \((c,0)\).
2. Look for a common factor or familiar pattern. Try a greatest common factor, a difference of squares or factoring by grouping.
3. Solve every factor equation. If a product is zero, at least one factor must equal zero. Keep repeated factors when describing multiplicity.
4. Use another method when factoring is difficult. For \(ax^2+bx+c=0\), with \(a\ne0\), the quadratic formula is \(x=(-b\pm\sqrt{b^2-4ac})/(2a)\). Graphing or numerical methods can estimate real roots.
5. Check in the original polynomial. Substitute each proposed zero. A rounded numerical approximation may leave a small nonzero residual.

For integer-coefficient polynomials, the Rational Zero Theorem supplies possible rational roots from factors of the constant and leading coefficients. Each candidate must be tested. Synthetic division by \(x-c\) gives remainder \(f(c)\); a zero remainder confirms the factor. Neither procedure guarantees that every root is rational.

A nonzero polynomial of degree \(n\) has at most \(n\) distinct real zeros. Over the complex numbers it has exactly \(n\) zeros counted with multiplicity. For real coefficients, nonreal roots occur in conjugate pairs; that fact helps check a result but does not find all roots by itself.

Cubic example: factor before solving

Find the zeros of \(f(x)=x^3-12x^2+20x\).

1. Take out the common factor: \(f(x)=x(x^2-12x+20)\).
2. Find two numbers with product 20 and sum −12: −2 and −10. Thus \(f(x)=x(x-2)(x-10)\).
3. Set the factors equal to zero: \(x=0\), \(x-2=0\), or \(x-10=0\).
4. The zeros are \(0,2,10\). Substituting any of these into the original polynomial gives 0.

Do not divide the equation by \(x\) without first considering \(x=0\). Doing so would discard one of the solutions.

Real intercepts, repeated zeros and complex roots

The graph of a real polynomial crosses the x-axis at a real zero of odd multiplicity and touches it without crossing at a real zero of even multiplicity. For example, \((x-5)^2\) has one distinct zero, 5, with multiplicity 2. Its graph touches the axis at \((5,0)\).

A graph can help estimate real roots, but a viewing window can miss an intercept or hide a repeated zero. Nonreal roots are not intercepts on an ordinary real x-y graph: \(x^2+1=0\) has roots \(i\) and \(-i\), while the real graph has no x-intercepts.

Practice: find the zeros

Factor each polynomial and identify its distinct zeros. Note any repeated root.

1. \(3x^3-19x^2+33x-9\)
2. \(x^2-10x+25\)
3. \(x^3+2x^2-25x-50\)
4. \(x^4+2x^3-16x^2-32x\)

Answers and factor checks

1. \((x-3)^2(3x-1)\): zeros 3 and \(1/3\); 3 has multiplicity 2.
2. \((x-5)^2\): zero 5 has multiplicity 2.
3. \((x+2)(x-5)(x+5)\): zeros −2, 5, −5.
4. \(x(x+2)(x-4)(x+4)\): zeros 0, −2, 4, −4.

Expand the factors to check each polynomial, then substitute the roots to check the zero equations.

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Continue with the Algebra 2 learning center. See the OpenStax polynomial-zero lesson and graph and multiplicity guidance for further explanation.