Lesson Worksheet: Rational Zeros Theorem Mathematics • 10th Grade

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?

  • A56, 56
  • B15, 15
  • C1, 1
  • D65, 65
  • E16, 16

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

  • A2, 15, 3
  • B2, 35, 1
  • C2, 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 (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.

  • 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
  • B13
  • C34
  • D23
  • E14

Q6:

True or False: If 𝑓(𝑥)=2𝑥12𝑥5𝑥+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+3𝑥8𝑥+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).

  • A1, 2, and 13
  • B1, 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+5𝑥25𝑥10𝑥+24.

  • A1, 2, 23, and 43
  • B1, 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)=4𝑥4𝑥13𝑥+7𝑥+6.

  • A1,2,4,16,13,23,431,2,4,16,13,23,43
  • B1,2,3,6,14,12,34,321,2,3,6,14,12,34,32
  • C1,2,3,12,321,2,3,12,32
  • D1,2,3,4,6,16,431,2,3,4,6,16,43
  • E1,2,13,12,231,2,13,12,23

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