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(π‘₯)=5π‘₯βˆ’6π‘₯βˆ’29π‘₯+6.

What is the smallest possible rational zero?

What are the two possible rational zeros closest to zero?

  • Aβˆ’56, 56
  • Bβˆ’15, 15
  • Cβˆ’1, 1
  • Dβˆ’65, 65
  • Eβˆ’16, 16

What are the rational zeros, if any, of 𝑔(π‘₯)?

  • Aβˆ’2, 15, 3
  • B2, 35, βˆ’1
  • Cβˆ’2, βˆ’35, 1
  • D2, βˆ’15, βˆ’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 β„Ž(π‘₯)=9π‘₯+6π‘₯βˆ’2π‘₯+68π‘₯βˆ’27π‘₯βˆ’14οŠͺ?

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 π‘˜(π‘₯)=9π‘₯βˆ’18π‘₯+35π‘₯βˆ’18?

Q4:

The cross section of a skate banister, shown in the diagram, can be modeled with the polynomial function β„Ž(𝑑)=βˆ’π‘‘+52π‘‘βˆ’74𝑑+78, where β„Ž 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 𝐡, given that point 𝐡 is 12 m above the ground.

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

Q5:

If 𝑓(π‘₯)=2π‘₯+π‘Žπ‘₯+𝑏π‘₯+𝑐π‘₯+12οŠͺ, 𝑧 is a zero of 𝑓(π‘₯), and π‘Ž, 𝑏, and 𝑐 are integer numbers, which of the following can be the value of 𝑧?

  • A32
  • Bβˆ’13
  • Cβˆ’34
  • D23
  • Eβˆ’14

Q6:

True or False: If 𝑓(π‘₯)=2π‘₯βˆ’12π‘₯βˆ’5π‘₯+8π‘₯+3οŠͺ and 𝑧 is a zero of 𝑓(π‘₯), then 𝑧 can be found using the rational zeros theorem.

  • AFalse
  • BTrue

Q7:

Consider the rational zeros theorem for the function 𝑓(π‘₯)=2π‘₯+3π‘₯βˆ’8π‘₯+3.

Determine all factors of the constant term of 𝑓(π‘₯).

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

Determine all factors of the leading coefficient of 𝑓(π‘₯).

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

Determine all possible rational zeros of 𝑓(π‘₯).

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

Determine the rational zeros of 𝑓(π‘₯).

  • Aβˆ’1, βˆ’2, and 13
  • Bβˆ’1, 3, and βˆ’12
  • C1, βˆ’3, and 12
  • D1, 3, and βˆ’12
  • E1, βˆ’2, and 13

Q8:

True or False: If 𝑓(π‘₯) is a polynomial with integer coefficients and 2 is a zero of 𝑓(π‘₯), 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(π‘₯)=6π‘₯+5π‘₯βˆ’25π‘₯βˆ’10π‘₯+24οŠͺ.

  • Aβˆ’1, 2, βˆ’23, and 43
  • Bβˆ’1, 2, 23, and βˆ’43
  • C1, βˆ’2, 23, and βˆ’34
  • D1, βˆ’2, 32, and 34
  • E1, βˆ’2, 32, and βˆ’43

Q10:

Using the rational zeros theorem, list all the possible rational zeros of the function 𝑓(π‘₯)=4π‘₯βˆ’4π‘₯βˆ’13π‘₯+7π‘₯+6οŠͺ.

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

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