1 Chapter 5 Test Review

Pre-AP Algebra II – Chapter 5 Test Review Standards/Goals:

A.1.b./A.APR.5.: o I can expand a binomial using Pascal’s Triangle. o I can use the binomial theorem to expand a binomial.

A.1.c./F.1.b.: I can factor a quadratic trinomial in the form of 𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄 A.1.c./A.SSE.1.a.:

o I can interpret parts of an expression, such as terms, factors, and coefficients o I can factor a quadratic expression

B.1.c./F.IF.5.: o I can fit real world data to linear, quadratic, cubic, or quartic models. o I can relate the domain to a function and describe a quantitative relationship.

E.1.b./A.APR.3.: o I can identify zeros of polynomials when suitable factorizations are available. o I can solve a quadratic equation and identify the zeros using graphing and/or factoring methods.

E.1.c./N.CN.7.: o I can solve quadratic equations with real coefficients that have complex solutions. o I can use the Conjugate Root Theorem to solve equations.

E.2.b.: I can use transformations to consider the graph of a polynomial function (cubic). F.1.a./F.IF.7.c/A.APR.1.:

o I can identify the degree of a polynomial function. o I can evaluate and simplify polynomial expressions and equations.

F.1.b./A.SSE.2.a.: o I can write polynomials in factored form. o I can write a quadratic expression as a product of two binomials. o I can use the structure of an expression to identify ways to rewrite it.

F.1.b./A.APR.2.: o I can use long division to divide two polynomials. o I can use synthetic division to divide two polynomials.

F.1.b./ A.SSE.2.: I can factor polynomials using a variety of methods (factor theorem, synthetic division, long division, sums and differences of cubes, grouping)

F.2.a./A.APR.2.: o I can determine the number and type of rational zeros for a polynomial function. o I can use the Rational Root Theorem to solve equations.

F.2.b./A.APR.3.: I can find all rational zeros of a polynomial function. F.2.c./F.IF.7.c./A.REI.11.: :

o I can use the graph of a polynomial function to find the zeros/solutions. o I can understand the relationship (and connection) between zeros, roots, solutions and x-intercepts

of polynomial equations. o I can find the multiplicity of the graph of a polynomial. o I can solve a polynomial equation using factoring. o I can recognize the connection among zeros of a polynomial function, x-intercepts, factors of

polynomials, and solutions of polynomial equations. F.2.d./F.IF.7./F.IF.4.: I can use technology to graph a polynomial function and approximate the zeros,

intervals in which the function is increasing and/or decreasing (and constant) the minimum and maximum values and determine the domain and range of a polynomial function.

F.2.d./N.CN.8.: I can solve polynomial equation using the Fundamental Theorem of Algebra.

#1. Find all solutions of the equation: 𝑥4 + 2𝑥3 + 4𝑥2 + 8𝑥 = 0.

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#2. Consider the factored polynomial function: f(x) = (𝟐𝒙 − 𝟓)𝟐(𝒙 + 𝟐) Identify the zeros of this function and state the multiplicity of each.

a. _____________ is a zero of multiplicity _____________.

b. _____________ is a zero of multiplicity _____________. c. State whether the graph crosses the x-axis or touches and turns around at each zero.

#3. Determine the degree, describe the behavior of the graph of the polynomial function as 𝑥 → ±∞

and identify the x and y intercepts: f(x) = −(𝒙 + 𝟏)𝟐(𝒙 − 𝟐)(𝒙 + 𝟑) ANSWERS: SHOW ALL WORK HERE:

Degree: ________ Behavior: As x → ∞ f(x) → ________ As x → −∞ f(x) → ________

X-Intercepts:

Y-Intercepts:

#4. Use all available methods (in particular, the Conjugate Roots Theorem, if applicable) to factor the following polynomial equation completely, making use of the given zero. f(x) = 𝒙𝟒 − 𝟏𝟒𝒙𝟑 + 𝟗𝟖𝒙𝟐 − 𝟔𝟖𝟔𝒙 + 𝟐𝟒𝟎𝟏; 𝟕𝒊 is a zero Show work here:

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#5. What are the real roots of: 𝑥3 + 27 = 0?

#6. What is the factored form of: 2𝑥3 + 2𝑥2 − 40𝑥?

#7. What are the zeros of: y = (x + 9)(4x + 2)(x – 8)?

#8. What is the multiplicity of f(x) = (𝑥 − 6)7?

#9. For this polynomial: 𝑦 = (𝑥 + 9)4, what type of behavior will the graph exhibit at the x-

intercept?

#10. A quintic polynomial has rational coefficients. If √6 and 10 – i are roots of P(x) = 0, what

are the additional roots?

#11. Factor the following and determine the zeros and the multiplicities.

𝑓(𝑥) = 𝑥3 + 10𝑥2 + 25𝑥

#12. A polynomial with real coefficients has roots of: 7, -8, 10i, and −√5. What must be some

other ROOTS of this polynomial?

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#13. The following is the graph of a cubic function. How many ‘distinct’ zeros does it have?

How many zeros are ‘real’ and how many are complex?

#14. Evaluate this function for x = -2: f(x) = 6𝑥3 − 𝑥2 + 4𝑥 − 8.

#15. A fourth degree polynomial, P(x) with real coefficients has 4 distinct zeros, 2 of them are

10 and 6 – i. What must be true about the other two?

#16. A fifth degree polynomial has been factored to be:

𝑓(𝑥) = (𝑥 + 3)(𝑥 − 8)2(𝑥2 + 4) . How many x-intercepts does it have?

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#17. What are the zeros of: 𝑓(𝑥) = (5𝑥 + 4)(𝑥2 − 9)(𝑥 − 10)?

#18. Factor: 3𝑥2 + 2𝑥 − 16

#19. The following polynomials have a degree of 5. How many ‘distinct’ real zeros are there for

each graph? How many are real and how many are complex/imaginary?

#1. #2. #3.

#20. What numbers are zeros and what are their multiplicities for:

𝑥3 + 8𝑥2 + 16𝑥?

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#21. F.2.c.: Consider the graph of the function: f(x) = 𝑥4 − 25𝑥2 − 36𝑥, which has one x-

intercept at (-4,0). Find all the other zeros of the function algebraically. Show your work, and

explain how you found all your answer. Additionally, name the intervals where the function

both INCREASES and DECREASES.

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#22. Consider the graph of the function shown below:

a. Write the intervals in which the function is INCREASING or DECREASING.

b. Where are the approximate zeros of this function?

c. Assuming that the turning points shown on this graph are the ONLY turning points for the entire polynomial, what would possibly be smallest potential DEGREE for this particular polynomial function?

d. Are there any zeros for this function that might possibly have a multiplicity greater than 1? If so, where at?

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What are the zeros for the following polynomial functions? Additionally, state the degree of each, and determine the # and types of zeros (real, imaginary, distinct). How many x-intercepts will each function have? #23. f(x) = x(x - 8)(𝑥2 + 4)2(𝑥2 − 81)

ANSWERS: List ALL of the Zeros:

Degree:

# of Real Zeros:

# of Imaginary Zeros:

# of Distinct REAL Zeros:

# of x-intercepts:

#24. f(x) = (𝑥 − 7)3(𝑥 + 4)(2x – 10)

ANSWERS: List ALL of the Zeros:

Degree:

# of Real Zeros:

# of Imaginary Zeros:

# of Distinct REAL Zeros:

# of x-intercepts:

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FLASHBACK: Solve, graph and write each in interval form: #1. 6|𝑥| − 8 < 28 #2. 5|𝑥| + 10 ≥ 40 #3. −2|𝑥| + 18 ≤ 10 #4. −3|𝑥| + 10 ≥ 19 #5. Consider the following quadratic: 𝑦 = 𝑥2 − 6𝑥 + 18 Determine the number and type of roots for this equation using the discriminant.

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Consider the following function: 𝑦 = −2𝑥2 + 12𝑥 + 9 #6. What is the vertex of the function?

#7. What is the axis of symmetry?

#8. Does the parabola have a minimum or a maximum and where at?

#9. Where is the y-intercept? Let m = 5 – 7i and h = 2 + 9i #10. Find m – 6h #11. Find m · h

#12. What is 𝑚

ℎ? #13. What is m + h?

Use the following functions to answer the questions: f(x) = 3x – 2 g(x) = 3𝑥2 + 2𝑥 − 1 h(x) = 4x + 8 k(x) = 2𝑥2 − 𝑥 − 9

#14. Find (𝑓

ℎ)(−2). #15. Find (g – k)(5)