# 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 .

- A
- B
- C
- D
- E

**Q2: **

Hannah is trying to find zeros in the function . She has used synthetic division to find for , , 1, and 3.

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

- A
- B
- C
- D
- E

**Q3: **

Consider the function .

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

What can he conclude about β5?

- 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

What can he conclude about 2?

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

Find all the real zeros of .

- A2, 13
- Bβ2, 2
- C1, 4
- Dβ13, 1
- Eβ4, 2

**Q4: **

Use synthetic division to determine whether 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 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 lie?

- A
- B
- C
- D
- E

**Q6: **

If is an upper bound for the real zeros of , which of the following can be ?

- A
- B
- C
- D
- E

**Q7: **

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

- A is an upper bound for the real zeros of .
- B is a lower bound for the real zeros of .
- C is a real zero of .
- DNone of the above

**Q8: **

If each number in the bottom row of the synthetic division of by is either positive or zero, where is a polynomial with real coefficients and a positive leading coefficient, which of the following must be true?

- A is a lower bound for the real zeros of .
- B is an upper bound for the real zeros of .
- C is a real zero of .
- DNone of the above

**Q9: **

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

- A is neither an upper bound nor a lower bound for the real zeros of .
- B is an upper bound for the real zeros of .
- C is a lower bound for the real zeros of .
- D is a real zero of .
- ENone of the above

**Q10: **

If each number in the bottom row of the synthetic division of by is either positive or zero, where is a polynomial with real coefficients and a positive leading coefficient, which of the following must be true?

- A is a real zero of .
- B is a lower bound for the real zeros of .
- C is an upper bound for the real zeros of .
- D is neither an upper bound nor a lower bound for the real zeros of .
- ENone of the above