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(π‘₯)=4π‘₯+π‘₯βˆ’27π‘₯+18π‘₯οŠͺ.

  • A3,0,βˆ’34,βˆ’2
  • Bβˆ’3,0,34,2
  • Cβˆ’3,0,βˆ’43,βˆ’2
  • D3,34,2
  • Eβˆ’3,0,43,2

Q2:

Hannah is trying to find zeros in the function 𝑓(π‘₯)=6π‘₯+19π‘₯βˆ’37π‘₯βˆ’62π‘₯+24οŠͺ. 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[βˆ’2,1]
  • B[1,3]
  • C[βˆ’2,3]
  • D[βˆ’5,3]
  • E[βˆ’5,1]

Q3:

Consider the function 𝑓(π‘₯)=π‘₯βˆ’4π‘₯βˆ’7π‘₯+74π‘₯βˆ’104οŠͺ.

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

What can he conclude about βˆ’5?

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

What can he conclude about 2?

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

Find all the real zeros of 𝑓.

  • A2, 13
  • Bβˆ’2, 2
  • C1, 4
  • Dβˆ’13, 1
  • Eβˆ’4, 2

Q4:

Use synthetic division to determine whether π‘₯=6 is an upper bound, a lower bound, or neither an upper nor a lower bound on the interval in which all real zeros of the polynomial function 𝑓(π‘₯)=π‘₯+3π‘₯βˆ’34π‘₯βˆ’42 lie.

  • AIt is a lower bound.
  • BIt is neither an upper nor a lower bound.
  • CIt is an upper bound.

Q5:

Which the following is the interval in which all real zeros of the polynomial function 𝑓(π‘₯)=π‘₯+9π‘₯+23π‘₯+15 lie?

  • A[βˆ’10,βˆ’2]
  • B[βˆ’9,βˆ’1]
  • C[βˆ’4,1]
  • D[βˆ’11,βˆ’3]
  • E[βˆ’2,2]

Q6:

If π‘₯=1 is an upper bound for the real zeros of 𝑓(π‘₯), which of the following can be 𝑓(π‘₯)?

  • A𝑓(π‘₯)=π‘₯βˆ’4π‘₯+π‘₯+6
  • B𝑓(π‘₯)=6π‘₯+19π‘₯+2π‘₯βˆ’3
  • C𝑓(π‘₯)=2π‘₯βˆ’13π‘₯+17π‘₯+12
  • D𝑓(π‘₯)=π‘₯βˆ’5π‘₯+2π‘₯+8
  • E𝑓(π‘₯)=3π‘₯βˆ’2π‘₯βˆ’7π‘₯βˆ’2

Q7:

If the numbers in the bottom row of the synthetic division of 𝑓(π‘₯) by π‘₯βˆ’3 are alternately positive and negative (zero entries count as positive or negative), where 𝑓(π‘₯) is a polynomial with real coefficients and a positive leading coefficient, which of the following must be true?

  • Aπ‘₯=3 is an upper bound for the real zeros of 𝑓.
  • Bπ‘₯=3 is a lower bound for the real zeros of 𝑓.
  • Cπ‘₯=3 is a real zero of 𝑓.
  • DNone of the above

Q8:

If each number in the bottom row of the synthetic division of 𝑓(π‘₯) by π‘₯+1 is either positive or zero, where 𝑓(π‘₯) is a polynomial with real coefficients and a positive leading coefficient, which of the following must be true?

  • Aπ‘₯=βˆ’1 is a lower bound for the real zeros of 𝑓.
  • Bπ‘₯=βˆ’1 is an upper bound for the real zeros of 𝑓.
  • Cπ‘₯=βˆ’1 is a real zero of 𝑓.
  • DNone of the above

Q9:

If the numbers in the bottom row of the synthetic division of 𝑓(π‘₯) by π‘₯+3 are alternately positive and negative (zero entries count as positive or negative), where 𝑓(π‘₯) is a polynomial with real coefficients and a positive leading coefficient, which of the following must be true?

  • Aπ‘₯=βˆ’3 is neither an upper bound nor a lower bound for the real zeros of 𝑓.
  • Bπ‘₯=βˆ’3 is an upper bound for the real zeros of 𝑓.
  • Cπ‘₯=βˆ’3 is a lower bound for the real zeros of 𝑓.
  • Dπ‘₯=βˆ’3 is a real zero of 𝑓.
  • ENone of the above

Q10:

If each number in the bottom row of the synthetic division of 𝑓(π‘₯) by π‘₯βˆ’1 is either positive or zero, where 𝑓(π‘₯) is a polynomial with real coefficients and a positive leading coefficient, which of the following must be true?

  • Aπ‘₯=1 is a real zero of 𝑓.
  • Bπ‘₯=1 is a lower bound for the real zeros of 𝑓.
  • Cπ‘₯=1 is an upper bound for the real zeros of 𝑓.
  • Dπ‘₯=1 is neither an upper bound nor a lower bound for the real zeros of 𝑓.
  • ENone of the above

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