Worksheet: Upper and Lower Bound Tests for Polynomial Functions

In this worksheet, we will practice using upper and lower bound tests to verify if the given interval is the interval that contains all real zeros.

Q1:

Using synthetic division and the upper and lower bound tests, find all real zeros of the function 𝑓 ( 𝑥 ) = 4 𝑥 + 𝑥 2 7 𝑥 + 1 8 𝑥 4 3 2 .

  • A 3 , 0 , 3 4 , 2
  • B 3 , 3 4 , 2
  • C 3 , 0 , 4 3 , 2
  • D 3 , 0 , 3 4 , 2
  • E 3 , 0 , 4 3 , 2

Q2:

Rania is trying to find zeros in the function 𝑓 ( 𝑥 ) = 6 𝑥 + 1 9 𝑥 3 7 𝑥 6 2 𝑥 + 2 4 . She has used synthetic division to find 𝑓 ( 𝑎 ) for 𝑎 = 5 , 2 , 1, and 3.

Use her results to state an interval in which all real zeros of 𝑓 lie.

  • A [ 1 , 3 ]
  • B [ 5 , 1 ]
  • C [ 2 , 1 ]
  • D [ 5 , 3 ]
  • E [ 2 , 3 ]

Q3:

Consider the function 𝑓 ( 𝑥 ) = 𝑥 4 𝑥 7 𝑥 + 7 4 𝑥 1 0 4 4 3 2 .

William is using synthetic division to help him find real zeros of 𝑓 .

What can he conclude about −5?

  • AThat it is a real zero of 𝑓
  • BThat it is an upper bound on the interval in which all real zeros lie
  • CThat it is neither upper bound nor lower bound on the interval in which all real zeros lie
  • DThat it is a lower bound on the interval in which all real zeros lie

What can he conclude about 2?

  • AThat it is a real zero of 𝑓
  • BThat it is the only real zero of 𝑓
  • CThat it is a lower bound on the interval in which all real zeros lie

Find all the real zeros of 𝑓 .

  • A−2, 2
  • B1, 4
  • C−4, 2
  • D2, 13
  • E−13, 1

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