Lesson Worksheet: Rational Zeros Theorem Mathematics • 10th Grade

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In this worksheet, we will practice using the rational zeros theorem to list all possible rational zeros in a given polynomial function.

Q1:

Using the rational zeros theorem, you can list all the possible rational zeros of the function 𝑔(𝑥)=5𝑥−6𝑥−29𝑥+6.

What is the smallest possible rational zero?

What are the two possible rational zeros closest to zero?

  • A−56, 56
  • B−15, 15
  • C−1, 1
  • D−65, 65
  • E−16, 16

What are the rational zeros, if any, of 𝑔(𝑥)?

  • A−2, 15, 3
  • B2, 35, −1
  • C−2, −35, 1
  • D2, −15, −3

Q2:

The rational zeros theorem can be used to generate a list of all possible rational zeros of a polynomial which we can then check one by one. How many possible rational zeros does the rational zeros theorem give us for the function ℎ(𝑥)=9𝑥+6𝑥−2𝑥+68𝑥−27𝑥−14?

Q3:

The rational zeros theorem can be used to generate a list of all possible rational zeros of a polynomial which we can then check one by one. How many possible rational zeros does the rational zeros theorem give us for the function 𝑘(𝑥)=9𝑥−18𝑥+35𝑥−18?

Q4:

The cross section of a skate banister, shown in the diagram, can be modeled with the polynomial function ℎ(𝑑)=−𝑑+52𝑑−74𝑑+78, where ℎ is the height above the ground and 𝑑 is the horizontal distance from point 𝐴.

Using the rational zeros theorem or otherwise, determine the horizontal distance between point 𝐴 and point 𝐵, given that point 𝐵 is 12 m above the ground.

  • A3 m
  • B12 m
  • C14 m
  • D1 m
  • E18 m

Q5:

If 𝑓(𝑥)=2𝑥+𝑎𝑥+𝑏𝑥+𝑐𝑥+12, 𝑧 is a zero of 𝑓(𝑥), and 𝑎, 𝑏, and 𝑐 are integer numbers, which of the following can be the value of 𝑧?

  • A32
  • B−13
  • C−34
  • D23
  • E−14

Q6:

True or False: If 𝑓(𝑥)=2𝑥−12𝑥−5𝑥+8𝑥+3 and 𝑧 is a zero of 𝑓(𝑥), then 𝑧 can be found using the rational zeros theorem.

  • AFalse
  • BTrue

Q7:

Consider the rational zeros theorem for the function 𝑓(𝑥)=2𝑥+3𝑥−8𝑥+3.

Determine all factors of the constant term of 𝑓(𝑥).

  • A+3 and −3
  • B+1, −1, +3, and −3
  • C+2, −2, +3, and −3
  • D+1 and −1
  • E+2 and −2

Determine all factors of the leading coefficient of 𝑓(𝑥).

  • A+2, −2, +3, and −3
  • B+1 and −1
  • C+2 and −2
  • D+3 and −3
  • E+1, −1, +2, and −2

Determine all possible rational zeros of 𝑓(𝑥).

  • A+12, −12, +32, −32, +3, and −3
  • B+2, −2, +3, and −3
  • C+1, −1, +12, −12, +32, −32, +3, and −3
  • D+1, −1, +12, −12, +32, and −32
  • E+1, −1, +2, −2, +3, and −3

Determine the rational zeros of 𝑓(𝑥).

  • A−1, −2, and 13
  • B−1, 3, and −12
  • C1, −3, and 12
  • D1, 3, and −12
  • E1, −2, and 13

Q8:

True or False: If 𝑓(𝑥) is a polynomial with integer coefficients and 2 is a zero of 𝑓(𝑥), then this zero can be found using the rational zeros theroem.

  • AFalse
  • BTrue

Q9:

Use the rational zeros theorem to find the rational zeros of 𝑓(𝑥)=6𝑥+5𝑥−25𝑥−10𝑥+24.

  • A−1, 2, −23, and 43
  • B−1, 2, 23, and −43
  • C1, −2, 23, and −34
  • D1, −2, 32, and 34
  • E1, −2, 32, and −43

Q10:

Using the rational zeros theorem, list all the possible rational zeros of the function 𝑓(𝑥)=4𝑥−4𝑥−13𝑥+7𝑥+6.

  • A1,2,4,16,13,23,43−1,−2,−4,−16,−13,−23,−43
  • B1,2,3,6,14,12,34,32−1,−2,−3,−6,−14,−12,−34,−32
  • C1,2,3,12,32−1,−2,−3,−12,−32
  • D1,2,3,4,6,16,43−1,−2,−3,−4,−6,−16,−43
  • E1,2,13,12,23−1,−2,−13,−12,−23

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