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Worksheet: Upper and Lower Bound Tests for Polynomial Functions

Q1:

Hannah is trying to find zeros in the function 𝑓 ( π‘₯ ) = 6 π‘₯ + 1 9 π‘₯ βˆ’ 3 7 π‘₯ βˆ’ 6 2 π‘₯ + 2 4 4 3 2 . She has used synthetic division to find 𝑓 ( π‘Ž ) for π‘Ž = βˆ’ 5 , βˆ’ 2 , 1, and 3.

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

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

Q2:

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

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

Q3:

Consider the function 𝑓 ( π‘₯ ) = π‘₯ βˆ’ 4 π‘₯ βˆ’ 7 π‘₯ + 7 4 π‘₯ βˆ’ 1 0 4 4 3 2 .

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

What can he conclude about βˆ’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 .

  • Aβˆ’2, 2
  • B1, 4
  • Cβˆ’4, 2
  • D2, 13
  • Eβˆ’13, 1