Pre-Calculus Unit 2 Test Review Name ________________________

Rationalizing Denominators: Know how to rationalize and simplify fractions. Show all work to receive full credit.

Quadratics: Converting Standard Form to Vertex Form (and vice versa); Solving (Quadratic Formula, Factoring, & Square Root Method); Applications (Root or Vertex Word Problems → Optimization)

Convert from Standard Form to Vertex Form. Show all work to receive full credit.

Convert from Vertex Form to Standard Form. Show all work to receive full credit.

Choose a method of solving (quadratic formula, factoring, square roots) for the following quadratic equations. State all answers in SIMPLIFIED, EXACT FORM. Show all work to receive full credit.

a. 𝑥2 + 10𝑥 + 29 = 0

Method: ________________________

Answer: _________________

b. 𝑥2 − 3𝑥 = 18

Method: ________________________

Answer:_________________

c. 6𝑥2 + 384 = 0

Method: ________________________

Answer:_________________

d. 3𝑥2 + 6𝑥 − 2 = 0

Method: ________________________

Answer: _________________

Quadratic Applications:

1. A farmer wants to build a rectangular garden bed for the upcoming harvest season. He only has enough materials to build a bed with a perimeter of 42 yards. The farmer makes a profit of $18 per square yard.

A) What are the dimensions that will produce the maximum area of such a rectangle, so that the farmer can optimize his profits? Show all work to receive full credit.

B) What is the maximum area of the garden bed? Show all work to receive full credit.

C) What is the maximum profit the farmer can make for his new garden bed? Show all work to receive full credit.

2. The perimeter of a rectangular yard is 70 yards. What are the dimensions that will produce the maximum area? You will only be given credit for algebraic solutions!

3. The sum of two numbers is 18. What is the maximum value of their product? You will only be given credit for algebraic solutions!

4. The product of 2 consecutive positive integers is 25 more than the first. Find the integers. You will only be given credit for algebraic solutions!

**Also, understand what would change is I asked for 2 consecutive positive odd/even integers!!**

Polynomials: Identifying Polynomial or Not; Characteristics of Polynomials (Max/Min, Real Zeros, End Behavior, Domain, Range, Intervals of Increasing/Decreasing); Finding Zeros/Multiplicity; Writing Polynomials from Zeros; Remainder Theorem; Factor Theorem; Fundamental Theorem of Algebra; Rational Zero Theorem (The P/Q Method)

1. Determine if each example is a polynomial or not (know what makes something NOT a polynomial!!):

2. Label the feature with the indicated letter (Not all letters will be used and some will be used more than once!): Global Max: A Global Min: B Local Max: C Local Min: D Inflection Point: E

3. ℎ(𝑥) = 𝑥3 + 𝑥2 − 4𝑥 − 4

Sketch of graph. **Round any decimal answers to the nearest hundredth! **

Maximum points (relative or absolute?):

Minimum point(s) (relative or absolute?):

REAL zeros:

End behavior:

Domain:

Range:

Intervals of increasing:

Intervals of decreasing:

Example Polynomial? 3𝑥 √𝑥

−6𝑦2 −79

𝑥

3𝑥 + 2

5 3𝑥𝑦−2 𝑥 − 2

4𝑥2 + 2𝑥1/2

4. Use the graph at right to identify the following:

a) Roots (Note any Multiplicity):

b) End Behavior (in correction notation):

c) Sign of Leading Coefficient Explain how you know!

d) Possible Equation (leave in factored form)

5. What are the roots of ℎ(𝑥) = 𝑥3 + 9𝑥2 + 20𝑥? Show all work to receive full credit.

6. A polynomial has roots 0, -1, and -4 (multiplicity of 2). Write an equation for this polynomial.

7. A polynomial has roots −5 and 3𝑖. Write a possible equation of this function in standard form. Show all work to receive full credit.

8. What is the Remainder Theorem?

9. What is the Factor Theorem?

10. What is the Fundamental Theorem of Algebra?

11. What is the remainder when (2x3 + 62x - 26x2 + 4) is divided by (2x - 6)?

12. Is (x - 4) is a factor of 𝑥4 + 2𝑥3 − 3𝑥 − 4? Explain how you know.

13. Find all possible rational roots of the function 𝑓(𝑥) = 2𝑥3 + 3𝑥 − 5

14. Find all roots (real and imaginary) of 𝑓(𝑥) = 6𝑥3 + 11𝑥2 − 3𝑥 − 2.

“Bonus Material”- Advanced Quadratic Word Problems; “Fancy Factoring”; Advanced Understanding of the Remainder/Factor Theorems; Advanced Rationalizing of Denominators

Good Luck! ☺