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Polynomial Zeros Practice Problems

Finding zeros is really finding factors, so these twelve problems work through the routes that get a polynomial into factored form: reading a factored expression directly, grouping, differences of squares, and substitution on a quartic. Multiplicity questions are mixed in, because a zero with an even multiplicity behaves completely differently on a graph.

  1. 1.

    Find all zeros of .

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    A product is zero when any factor is zero. gives , and gives . The sign always flips from the factor to the zero.

  2. 2.

    Find all zeros of .

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    This is a difference of squares: , so the zeros are and .

  3. 3.

    What is the multiplicity of the zero in ?

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    Multiplicity is the exponent on that factor. appears to the power 3, so the zero at has multiplicity 3. Being odd, the graph crosses the axis there — flattening as it goes.

  4. 4.

    Find all zeros of .

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    The bare is a factor too, so is a zero. The others come from and , giving and . Forgetting the zero contributed by a lone is a common slip.

  5. 5.

    For , at which zero does the graph touch the axis rather than cross it?

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    The zero at has multiplicity 2, which is even, so the curve comes down to the axis and turns back without passing through. The zero at has multiplicity 1 and is crossed normally.

  6. 6.

    Find all zeros of .

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    Take out the common factor first: . The zeros are . Dividing through by instead of factoring it out would lose the zero at the origin.

  7. 7.

    Find all zeros of .

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    Group in pairs: . The second bracket is a difference of squares, giving and zeros .

  8. 8.

    What do the multiplicities of add up to?

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    . That total is always the degree of the polynomial — every factor contributes its exponent to the degree, which is why counting zeros with multiplicity always reaches .

  9. 9.

    Find all real zeros of .

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    Substitute , turning it into . So or , giving and . Four zeros for a quartic — the maximum possible.

  10. 10.

    Find all zeros of .

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    Group: . The zeros are . A leading coefficient other than 1 is exactly what produces a fractional zero.

  11. 11.

    Find all zeros of .

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    Group: , giving .

  12. 12.

    What is the degree of ?

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    Add the degrees of the factors: . Note that contributes 2 to the degree but no real zeros at all — it is irreducible over the reals, so this quintic has only three real zeros.
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Polynomial Zeros Practice Problems

The finding polynomial zeros lesson works through every route from a polynomial to its factored form. These twelve problems drill those routes and mix in multiplicity questions, because multiplicity is what turns a list of zeros into a shape.

The one idea underneath all of it

Once a polynomial is written this way, the zeros are readable straight off — one per factor, with the exponent giving its multiplicity. Every technique in this set is really just a way of reaching that form.

The routes to factored form

SituationRoute
Already factoredread the zeros off, flipping each sign
Common factor in every termtake it out first
Four termsgroup in pairs
difference of squares
Only even powerssubstitute
None of the aboverational root theorem, then synthetic division

Multiplicity and what the graph does

A multiplicity of 1 crosses straight through. A multiplicity of 2 touches the axis and bounces back. A multiplicity of 3 still crosses, but flattens noticeably as it does. That distinction is what the graphing practice builds on.

Frequently Asked Questions

What is the difference between a zero, a root and an x-intercept?+

For a polynomial they are the same value described three ways. A zero is where the function equals zero, a root is a solution of the equation P(x) = 0, and an x-intercept is where the graph meets the axis. The words differ by context, not by meaning.

What does multiplicity mean?+

The exponent on a factor. In (x − 3)²(x + 1), the zero at 3 has multiplicity 2 and the zero at −1 has multiplicity 1. Multiplicity controls what the graph does at that point: odd crosses, even touches and turns back.

Why do the multiplicities add up to the degree?+

Because each factor contributes its exponent to the degree when the polynomial is multiplied out. This is the fundamental theorem of algebra in practice — a degree-n polynomial has exactly n zeros counted with multiplicity, once complex ones are included.

How do I find zeros when the polynomial will not factor easily?+

Use the rational root theorem to build a shortlist of candidates, test them with synthetic division, and divide out each hit. Once the remaining factor is a quadratic, switch to factoring or the quadratic formula rather than continuing to test.

Can a polynomial have fewer real zeros than its degree?+

Yes. Every irreducible quadratic factor contributes 2 to the degree and no real zeros — x² + 4 is the standard example. The count only reaches the degree exactly when complex zeros are included.

Why does factoring out an x matter so much?+

Because dividing by x throws away the zero at the origin. x³ − 4x has zeros at 0, 2 and −2; cancelling the x leaves only 2 and −2 and silently loses a third of the answer.

Calculators for this topic

Zeros Calculator

More practice problems

Rational Root Theorem Practice Problems End Behavior Practice Problems Graphing Polynomials Practice Problems