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Zeros of Rational Functions (College Board AP® Precalculus): Study Guide

Roger B

Written by: Roger B

Reviewed by: Mark Curtis

Updated on

Zeros of rational functions

Where do the real zeros of a rational function occur?

  • Let space r open parentheses x close parentheses equals fraction numerator p open parentheses x close parentheses over denominator q open parentheses x close parentheses end fraction be a rational function

    • i.e. where space p open parentheses x close parentheses and q open parentheses x close parentheses are both polynomial functions

  • The real zeros of bold italic r correspond to

    • real zeros of bold space bold italic p (the function in the numerator)

    • for such values as are also in the domain of bold italic r

  • What this means in practical terms is that the real zeros of r occur at values of xwhere

    • Error converting from MathML to accessible text.

    • but Error converting from MathML to accessible text.

      • Remember that any values of x must be excluded from the domain of r that make its denominator equal to zero

  • E.g. the function h open parentheses x close parentheses equals fraction numerator open parentheses x plus 4 close parentheses open parentheses x minus 3 close parentheses over denominator open parentheses x plus 1 close parentheses open parentheses x minus 3 close parentheses end fraction

    • The numerator is equal to zero when x equals negative 4 or x equals 3

      • But the denominator is also equal to zero when x equals 3

    • So space h has only one real zero, at x equals negative 4

What else do the real zeros of its numerator and denominator tell me about a rational function?

  • A rational function can only be equal to zero at one of its real zeros

    • I.e., these are the only points where the graph of the function can touch or cross the bold italic x-axis

  • However the real zeros of a rational function's numerator and denominator can help you determine other things about the behavior of the rational function

  • In particular, for a rational function space r open parentheses x close parentheses equals fraction numerator p open parentheses x close parentheses over denominator q open parentheses x close parentheses end fraction

    • The real zeros of space p and q

    • are the endpoints or asymptotes

    • for intervals satisfying the inequalities space r open parentheses x close parentheses greater or equal than 0 and space r open parentheses x close parentheses less or equal than 0

  • What this means in practical terms is this:

    • The real zeros of a rational function's numerator and denominator divide the function into a number of intervals

    • Within each of those intervals (i.e. not including any endpoints) the rational function is either positive everywhere or negative everywhere

      • So the numerator's and denominator's real zeros are the only points at which a rational function can change sign

      • (It may not change sign at such points, but it cannot change sign anywhere else)

    • And the rational function can only be equal to zero at the endpoint of such an interval which is also a real zero of the rational function

  • E.g. the function h open parentheses x close parentheses equals fraction numerator open parentheses x plus 4 close parentheses open parentheses x minus 3 close parentheses over denominator open parentheses x plus 1 close parentheses open parentheses x minus 3 close parentheses end fraction

    • The numerator is equal to zero when x equals negative 4 or x equals 3

      • and the denominator is equal to zero when x equals negative 1 or x equals 3

    • Those values of x, i.e. x equals negative 4, x equals negative 1 and x equals 3

      • divide the domain of h into four intervals

      • x less than negative 4, space minus 4 less than x less than negative 1, space minus 1 less than x less than 3 space and space x greater than 3

    • Test a value of x in the interval x less than negative 4

      • h open parentheses negative 5 close parentheses equals fraction numerator open parentheses open parentheses negative 5 close parentheses plus 4 close parentheses open parentheses open parentheses negative 5 close parentheses minus 3 close parentheses over denominator open parentheses open parentheses negative 5 close parentheses plus 1 close parentheses open parentheses open parentheses negative 5 close parentheses minus 3 close parentheses end fraction equals fraction numerator open parentheses negative 1 close parentheses open parentheses negative 8 close parentheses over denominator open parentheses negative 4 close parentheses open parentheses negative 8 close parentheses end fraction equals 1 fourth greater than 0

        • That is positive, so h open parentheses x close parentheses greater than 0 for all x less than negative 4

    • You can do the same thing for the other intervals

      • h open parentheses negative 2 close parentheses equals negative 2 less than 0, so h less than 0 for all x such that negative 4 less than x less than negative 1

      • h open parentheses 0 close parentheses equals 4 greater than 0, so h greater than 0 for all x such that negative 1 less than x less than 3

      • h open parentheses 4 close parentheses equals 8 over 5 greater than 0, so h greater than 0 for all x greater than 3

    • We have seen above that the only real zero of h is at x equals negative 4

    • Combining the above:

      • h open parentheses x close parentheses equals 0 at x equals negative 4

      • h open parentheses x close parentheses less than 0 on the interval negative 4 less than x less than negative 1

      • h open parentheses x close parentheses greater than 0 on the intervals space x less than negative 4, space minus 1 less than x less than 3, and space x greater than 3

Examiner Tips and Tricks

Considering real zeros of the numerator and denominator of a rational function in more detail can also allow you to identify the position of vertical asymptotes and holes for the function.

Worked Example

The function space f is given by space f open parentheses x close parentheses equals fraction numerator open parentheses x plus 3 close parentheses open parentheses x minus 1 close parentheses open parentheses x minus 5 close parentheses over denominator open parentheses x plus 1 close parentheses open parentheses x minus 1 close parentheses open parentheses x minus 3 close parentheses squared end fraction.

Without using a calculator:

(i)  find all values of x at which space f has a zero, or indicate there are no such values;

(ii)  determine all the intervals of x for which space f open parentheses x close parentheses greater than 0, and for which space f open parentheses x close parentheses less than 0.

Answer:

(i)

Zeros of a rational function can only occur at points where the numerator is equal to zero

open parentheses x plus 3 close parentheses open parentheses x minus 1 close parentheses open parentheses x minus 5 close parentheses equals 0

x equals negative 3 comma space space x equals 1 comma space space x equals 5

However those points will only be zeros of the rational function if the denominator is not also equal to zero

open parentheses x plus 1 close parentheses open parentheses x minus 1 close parentheses open parentheses x minus 3 close parentheses squared equals 0

x equals negative 1 comma space space x equals 1 comma space space x equals 3

Because the denominator is also equal to zero at x equals 1, that is not a zero of the rational function space f

The zeros of space f are at x equals negative 3 and x equals 5

(ii)

A rational function can only change sign at a point where its numerator or denominator is equal to zero

  • Those points were identified in part (i)

  • This means you must consider the following six intervals

x less than negative 3 comma space space minus 3 less than x less than negative 1 comma space space minus 1 less than x less than 1 comma space space 1 less than x less than 3 comma space space 3 less than x less than 5 space space and space space x greater than 5

Test the value of space f for a single point in each interval

  • For x less than negative 3

table row cell space f open parentheses negative 4 close parentheses end cell equals cell fraction numerator open parentheses open parentheses negative 4 close parentheses plus 3 close parentheses open parentheses open parentheses negative 4 close parentheses minus 1 close parentheses open parentheses open parentheses negative 4 close parentheses minus 5 close parentheses over denominator open parentheses open parentheses negative 4 close parentheses plus 1 close parentheses open parentheses open parentheses negative 4 close parentheses minus 1 close parentheses open parentheses open parentheses negative 4 close parentheses minus 3 close parentheses squared end fraction end cell row blank equals cell fraction numerator open parentheses negative 1 close parentheses open parentheses negative 5 close parentheses open parentheses negative 9 close parentheses over denominator open parentheses negative 3 close parentheses open parentheses negative 5 close parentheses open parentheses negative 7 close parentheses squared end fraction end cell row blank equals cell fraction numerator open parentheses negative 1 close parentheses open parentheses negative 5 close parentheses open parentheses negative 9 close parentheses over denominator open parentheses negative 3 close parentheses open parentheses negative 5 close parentheses open parentheses 49 close parentheses end fraction end cell row blank equals cell negative over positive end cell row blank less than 0 end table

  • For negative 3 less than x less than negative 1

table row cell space f open parentheses negative 2 close parentheses end cell equals cell fraction numerator open parentheses open parentheses negative 2 close parentheses plus 3 close parentheses open parentheses open parentheses negative 2 close parentheses minus 1 close parentheses open parentheses open parentheses negative 2 close parentheses minus 5 close parentheses over denominator open parentheses open parentheses negative 2 close parentheses plus 1 close parentheses open parentheses open parentheses negative 2 close parentheses minus 1 close parentheses open parentheses open parentheses negative 2 close parentheses minus 3 close parentheses squared end fraction end cell row blank equals cell fraction numerator open parentheses 1 close parentheses open parentheses negative 3 close parentheses open parentheses negative 7 close parentheses over denominator open parentheses negative 1 close parentheses open parentheses negative 3 close parentheses open parentheses negative 5 close parentheses squared end fraction end cell row blank equals cell fraction numerator open parentheses 1 close parentheses open parentheses negative 3 close parentheses open parentheses negative 7 close parentheses over denominator open parentheses negative 1 close parentheses open parentheses negative 3 close parentheses open parentheses 25 close parentheses end fraction end cell row blank equals cell 21 over 75 end cell row blank greater than 0 end table

  • For negative 1 less than x less than 1

table row cell space f open parentheses 0 close parentheses end cell equals cell fraction numerator open parentheses 0 plus 3 close parentheses open parentheses 0 minus 1 close parentheses open parentheses 0 minus 5 close parentheses over denominator open parentheses 0 plus 1 close parentheses open parentheses 0 minus 1 close parentheses open parentheses 0 minus 3 close parentheses squared end fraction end cell row blank equals cell fraction numerator open parentheses 3 close parentheses open parentheses negative 1 close parentheses open parentheses negative 5 close parentheses over denominator open parentheses 1 close parentheses open parentheses negative 1 close parentheses open parentheses negative 3 close parentheses squared end fraction end cell row blank equals cell fraction numerator open parentheses 3 close parentheses open parentheses negative 1 close parentheses open parentheses negative 5 close parentheses over denominator open parentheses 1 close parentheses open parentheses negative 1 close parentheses open parentheses 9 close parentheses end fraction end cell row blank equals cell fraction numerator 15 over denominator negative 9 end fraction end cell row blank less than 0 end table

  • For 1 less than x less than 3

table row cell space f open parentheses 2 close parentheses end cell equals cell fraction numerator open parentheses 2 plus 3 close parentheses open parentheses 2 minus 1 close parentheses open parentheses 2 minus 5 close parentheses over denominator open parentheses 2 plus 1 close parentheses open parentheses 2 minus 1 close parentheses open parentheses 2 minus 3 close parentheses squared end fraction end cell row blank equals cell fraction numerator open parentheses 5 close parentheses open parentheses 1 close parentheses open parentheses negative 3 close parentheses over denominator open parentheses 3 close parentheses open parentheses 1 close parentheses open parentheses negative 1 close parentheses squared end fraction end cell row blank equals cell fraction numerator open parentheses 5 close parentheses open parentheses 1 close parentheses open parentheses negative 3 close parentheses over denominator open parentheses 3 close parentheses open parentheses 1 close parentheses open parentheses 1 close parentheses end fraction end cell row blank equals cell fraction numerator negative 15 over denominator 3 end fraction end cell row blank less than 0 end table

  • For 3 less than x less than 5

table row cell space f open parentheses 4 close parentheses end cell equals cell fraction numerator open parentheses 4 plus 3 close parentheses open parentheses 4 minus 1 close parentheses open parentheses 4 minus 5 close parentheses over denominator open parentheses 4 plus 1 close parentheses open parentheses 4 minus 1 close parentheses open parentheses 4 minus 3 close parentheses squared end fraction end cell row blank equals cell fraction numerator open parentheses 7 close parentheses open parentheses 3 close parentheses open parentheses negative 1 close parentheses over denominator open parentheses 5 close parentheses open parentheses 3 close parentheses open parentheses 1 close parentheses squared end fraction end cell row blank equals cell fraction numerator open parentheses 7 close parentheses open parentheses 3 close parentheses open parentheses negative 1 close parentheses over denominator open parentheses 5 close parentheses open parentheses 3 close parentheses open parentheses 1 close parentheses end fraction end cell row blank equals cell fraction numerator negative 21 over denominator 15 end fraction end cell row blank less than 0 end table

  • For x greater than 5

table row cell space f open parentheses 6 close parentheses end cell equals cell fraction numerator open parentheses 6 plus 3 close parentheses open parentheses 6 minus 1 close parentheses open parentheses 6 minus 5 close parentheses over denominator open parentheses 6 plus 1 close parentheses open parentheses 6 minus 1 close parentheses open parentheses 6 minus 3 close parentheses squared end fraction end cell row blank equals cell fraction numerator open parentheses 9 close parentheses open parentheses 5 close parentheses open parentheses 1 close parentheses over denominator open parentheses 7 close parentheses open parentheses 5 close parentheses open parentheses 3 close parentheses squared end fraction end cell row blank equals cell fraction numerator open parentheses 9 close parentheses open parentheses 5 close parentheses open parentheses 1 close parentheses over denominator open parentheses 7 close parentheses open parentheses 5 close parentheses open parentheses 9 close parentheses end fraction end cell row blank equals cell positive over positive end cell row blank greater than 0 end table

Collect those results to write your final answer

space f open parentheses x close parentheses greater than 0 space in the intervals space minus 3 less than x less than negative 1 space and space x greater than 5

space f open parentheses x close parentheses less than 0 space in the intervals space x less than negative 3, space minus 1 less than x less than 1, space 1 less than x less than 3 space and space 3 less than x less than 5

Examiner Tips and Tricks

In part (ii) of the worked example, note that you don't actually care what the exact values of space f are at the various 'test points'.

  • You only care whether they are positive or negative

  • As shown for space f open parentheses negative 4 close parentheses and space f open parentheses 6 close parentheses, you can determine the sign of the function at a point without having to work out the exact value

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Author
Reviewer
Roger B

Author: Roger B

Expertise: Maths Content Creator

Roger's teaching experience stretches all the way back to 1992, and in that time he has taught students at all levels between Year 7 and university undergraduate. Having conducted and published postgraduate research into the mathematical theory behind quantum computing, he is more than confident in dealing with mathematics at any level the exam boards might throw at you.

Mark Curtis

Reviewer: Mark Curtis

Expertise: Maths Content Creator

Mark graduated twice from the University of Oxford: once in 2009 with a First in Mathematics, then again in 2013 with a PhD (DPhil) in Mathematics. He has had nine successful years as a secondary school teacher, specialising in A-Level Further Maths and running extension classes for Oxbridge Maths applicants. Alongside his teaching, he has written five internal textbooks, introduced new spiralling school curriculums and trained other Maths teachers through outreach programmes.