Worksheet: Rational Zeros Theorem

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)=5𝑥6𝑥29𝑥+6.

What is the smallest possible rational zero?

What are the two possible rational zeros closest to zero?

  • A−1, 1
  • B 1 6 , 1 6
  • C 6 5 , 6 5
  • D 5 6 , 5 6
  • E 1 5 , 1 5

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

  • A−2, 35, 1
  • B−2, 15, 3
  • C2, 15, −3
  • D2, 35, −1

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 (𝑥)=9𝑥+6𝑥2𝑥+68𝑥27𝑥14?

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)=9𝑥18𝑥+35𝑥18?

Q4:

The cross-section of a skate banister, shown in the diagram, can be modeled with the polynomial function (𝑑)=𝑑+52𝑑74𝑑+78, where 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.

  • A 1 4 m
  • B 1 2 m
  • C 1 m
  • D 1 8 m
  • E 3 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+0.3𝑥10.1𝑥+12.5𝑥2.1𝑥+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.

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