Lesson Worksheet: Rational Zeros Theorem Mathematics

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:

The value of an investment 𝐼 in a given start-up business is predicted to be given over time by the function 𝐼=𝐼𝑥+0.3𝑥−10.1𝑥+12.5𝑥−2.1𝑥+1, where 𝑥 is the number of decades after the investment.

The graph shows that the investment is predicted to first lose value. Given that it is predicted that the gain will be positive after less than a decade, determine how many years it will take for the investment to first return to the value of 𝐼.

Q6:

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

Q7:

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

Q8:

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

Q9:

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

Q10:

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

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.