Algebra 2 Unit 3 Lesson 4

Algebra 2 - Unit 3 Review - P1

By Henry Phan

​Which of the following is NOT a factor of f(x)

f(x) ​ = x³ + x² - 4x - 4

A. (x + ​2)

B. ​(x - 2)

C. (x - 3)

D. (x + 1​)

​Which of the following is NOT a factor of f(x)

f(x) ​ = x³ + x² - 4x - 4

A. (x + ​2)

B. ​(x - 2)

C. (x - 3)

D. (x + 1​)

​x ​ = -2, then f(x) = 0

x = -1, then f(x) = 0

x = 2​, then f(x) = 0

x = 3, ​f(x) = 20. Hence x - 3 is NOT a factor

​

​x ​ + 2, then x = - 2

x - 2, then x ​= 2

x -​ 3, then x = 3

x + 1​, then x = -1

Use the calculator to determine the value of x:

{-1, -2, 2}​

​

A​nswer C.​

Multiple Choice

Which of the following is NOT a factor of f(x)

f(x) ​ = x³ + 6x² - 7x - 60

• x + 5

• x + 4

• x + 3

• x - 3

Which of the following function has a factor of x + 1?

A. f(x) = x³ - 11x² + 20x + 33

​​B. f(x) = x³ - 13x² + 44x - 31

​C. f(x) = x³ - 13x² + 44x - 32

​D. f(x) = x³ - 11x² + 20x + 32​

Which of the following function has a factor of x - 2?

A. f(x) = x³ + 2x² - 9x - 18

​​B. f(x) = x³ + 3x² - 10x - 24

​C. f(x) = x³ + 9x² + 26x + 24

​D. f(x) = x³ + 5x² - 2x - 24

​x - 2 for x = 2

​A. Not b/c x = 2, f(x) = 20

B. Not b/c x = 2, f(x) = -24

C.​ Not b/c x = 2, f(x) = 120

D is correct answer. b/c x = 2 f(x) = 0

Multiple Choice

Which of the following function has a factor of x + 4?

• f(x) = x³ - x² + 14x + 22

• f(x) = x³ + x² - 14x + 23

• f(x) = x³ - x² - 14x + 24

• f(x) = x³ - x² - 14x - 25

​If f(x) = x³ + 3x² - 70x - 144 and x + 2 is a factor of f(x), then find all of the zeros of f(x) algebraically.

A. -9, -2​, 8

B. ​-9, -8, -2

C.​ -8, -2, 9

D.​ -2, 8, 9

​IIf f(x) = x³ + 3x² - 70x - 144 and x + 2 is a factor of f(x), then find all of the zeros of f(x) algebraically.

A. -9, -2​, 8

B. ​-9, -8, -2

C.​ -8, -2, 9

D.​ -2, 8, 9

​By a calculator:

​f(x) = x³ + 3x² - 70x - 144 given {-9, -2, 8}

​Answer A​

Multiple Choice

If f(x) = x⁴ + 2x³ - 43x² - 44x + 84 and (x - 1) is a factor of f(x), then find all of the zeros of f(x) algebraically.

• -4, 6, 7

• -7, -2, 6

• -2, 4, 9

• -6, 5, 7

​How many complex zero will the following function have?

F(x) =​ 4x³ + 3x² + x + 2

​A. 4

​B. 3

C. 2​

D. 1​

​How many complex zero will the following function have?

F(x) =​ 4x³ + 3x² + x + 2

​A. 4

​B. 3

C. 2​

D. 1​

​A number of complex zeros = highest degree

​

​Answer: B. 3

Multiple Choice

How many complex zero will the following function have?

F(x) = 2​x⁴ - x³ + 9x² + 33x + 65

• 4

• 3

• 2

• 1

Multiple Choice

How many complex zero will the following function have?

F(x) = 2​x⁴ - x³ + 9x² + 33x + 65 + 6x⁵ - 7x⁶

• 7

• 6

• 5

• 4

​Base on the graph below, which statement is false?

​A. 0 is a zero of the function with a multiplicity of 1

​B. The leading coefficient of the polynomial is negative.

C. The ​degree of the polynomial is even

D. 3​ is a root of the polynomial with a multiplicity of 1

​E. 5 is a root of the polynomial with a multiplicity of 2

​Base on the graph below, which statement is false?

​A. 0 is a zero of the function with a multiplicity of 1 (Yes)

​B. The leading coefficient of the polynomial is negative. (Yes)

C. The ​degree of the polynomial is even (Yes)

D. 3​ is a root of the polynomial with a multiplicity of 1 (Yes)

​E. 5 is a root of the polynomial with a multiplicity of 2 (No)

​x-interceps = roots/zeros = {-4, 0, 3, 5}

Multiple Choice

Base on the graph below, which statement is false?

• 5 is a root of the function with a multiplicity of 1

• 2 is a root of the polynomial with a multiplicity of 2

• -3​ is a root of the polynomial with a multiplicity of 2

• The leading coefficient of the polynomial is positive.

• The ​degree of the polynomial is even

​Based on the graph, the leading coefficient of the polynomial is _______, and the degree of the polynomial is ________

​Positive

​Negative

​Odd

​Even

​Based on the graph, the leading coefficient of the polynomial is _______, and the degree of the polynomial is ________

​Positive

​Negative

​Odd

​Even

​Based on the graph, the leading coefficient of the polynomial is _______, and the degree of the polynomial is ________

​Positive

​Negative

​Odd

​Even

​Based on the graph, the leading coefficient of the polynomial is _______, and the degree of the polynomial is ________

​Positive

​Negative

​Odd

​Even

Multiple Choice

Based on the graph, the leading coefficient of the polynomial is _______, and the degree of the polynomial is ________

• Leading coefficient = "positive"

  Degree = "odd"

• Leading coefficient = "negative"

  Degree = "odd"

• Leading coefficient = "positive"

  Degree = "even"

• Leading coefficient = "negative"

  Degree = "even"

Multiple Choice

Based on the graph, the leading coefficient of the polynomial is _______, and the degree of the polynomial is ________

• Leading coefficient = "positive"

  Degree = "odd"

• Leading coefficient = "negative"

  Degree = "odd"

• Leading coefficient = "positive"

  Degree = "even"

• Leading coefficient = "negative"

  Degree = "even"

​Based on the graph below, the real root are -6, -4, 4 and 2

​A. True

B.​ False

​Based on the graph below, the real root are -6, -4, 4, and 2

​A. True

B.​ False

​Zeros: {-6, -3, 2}

​Answer: B. False

Multiple Choice

Based on the graph below, the real root are -3, -2, and 1

• True

• False

​Determine the end behavior of the function:

F​(x) = x⁴ - 6x² + 2x + 5​

​

​

​

​F(x) →​ -∞, x →​ -∞​

F(x) →​ -∞, x →​ +∞​ ​

​F(x) →​ -∞, x →​ -∞​

F(x) →​ +∞, x →​ +∞​ ​

​F(x) →​ +∞, x →​ -∞​

F(x) →​ -∞, x →​ +∞​ ​

​F(x) →​ +∞, x →​ -∞​

F(x) →​ +∞, x →​ +∞​ ​

​A

B

C

D

​Determine the end behavior of the function:

F​(x) = x⁴ - 6x² + 2x + 5​​

​

​

​

​F(x) →​ -∞, x →​ -∞​

F(x) →​ -∞, x →​ -∞​ ​

​F(x) →​ -∞, x →​ -∞​

F(x) →​ +∞, x →​ +∞​ ​

​F(x) →​ +∞, x →​ -∞​

F(x) →​ -∞, x →​ +∞​ ​

​F(x) →​ +∞, x →​ -∞​

F(x) →​ +∞, x →​ +∞​ ​

​A

B

C

D

​Even - Positive

​

Answer: ​D

​Determine the end behavior of the function:

F​(x) = F(x) = x³ + 3x² - 7

​

​

​

​F(x) →​ -∞, x →​ -∞​

F(x) →​ -∞, x →​ +∞​ ​

​F(x) →​ -∞, x →​ -∞​

F(x) →​ +∞, x →​ +∞​ ​

​F(x) →​ +∞, x →​ -∞​

F(x) →​ -∞, x →​ +∞​ ​

​F(x) →​ +∞, x →​ -∞​

F(x) →​ +∞, x →​ +∞​ ​

​A

B

C

D

​Determine the end behavior of the function:

F​(x) = F(x) = x³ + 3x² - 7​

​

​

​

​F(x) →​ -∞, x →​ -∞​

F(x) →​ -∞, x →​ +∞​ ​

​F(x) →​ -∞, x →​ -∞​

F(x) →​ +∞, x →​ +∞​ ​

​F(x) →​ +∞, x →​ -∞​

F(x) →​ -∞, x →​ +∞​ ​

​F(x) →​ +∞, x →​ -∞​

F(x) →​ +∞, x →​ +∞​ ​

​A

B

C

D

​Odd - Positive

​

Answer: ​B

Multiple Choice

Determine the end behavior of the function:

F​(x) = -x⁶ - 4x⁵ + 3x + 4​

• $$f\left(x\right)\rightarrow-\infty\ as\ x\ \rightarrow-\infty$$   $$f\left(x\right)\rightarrow-\infty\ as\ x\rightarrow+\infty$$  

• $$f\left(x\right)\rightarrow-\infty\ as\ x\rightarrow\ -\infty$$   $$f\left(x\right)\rightarrow+\infty\ as\ x\rightarrow+\infty$$  

• $$f\left(x\right)\rightarrow+\infty\ as\ x\rightarrow\ -\infty$$   $$f\left(x\right)\rightarrow-\infty\ as\ x\rightarrow+\infty$$  

• $$f\left(x\right)\rightarrow+\infty\ as\ x\rightarrow-\infty$$   $$f\left(x\right)\rightarrow+\infty\ as\ x\rightarrow+\infty$$  

Multiple Choice

Determine the end behavior of the function:

F​(x) = -x⁴ - 4x⁵ + 3x + 4​

• $$f\left(x\right)\rightarrow-\infty\ as\ x\ \rightarrow-\infty$$   $$f\left(x\right)\rightarrow-\infty\ as\ x\rightarrow+\infty$$  

• $$f\left(x\right)\rightarrow-\infty\ as\ x\rightarrow\ -\infty$$   $$f\left(x\right)\rightarrow+\infty\ as\ x\rightarrow+\infty$$  

• $$f\left(x\right)\rightarrow+\infty\ as\ x\rightarrow\ -\infty$$   $$f\left(x\right)\rightarrow-\infty\ as\ x\rightarrow+\infty$$  

• $$f\left(x\right)\rightarrow+\infty\ as\ x\rightarrow-\infty$$   $$f\left(x\right)\rightarrow+\infty\ as\ x\rightarrow+\infty$$  

​What is the leading coefficient of the polynomial f(x) defined below?

f(x) = 8 + 9x - 7x³​

_______

​input value:

​What is the leading coefficient of the polynomial f(x) defined below?

f(x) = 8 + 9x - 7x³​

-7

​input value:

​What is the degree of the polynomial f(x) defined below?

f(x) = 9 - 8x³ - 6x² - x + x​⁴

_______

​input value:

​What is the degree of the polynomial f(x) defined below?

f(x) = 9 - 8x³ - 6x² - x + x​⁴

​input value:

4​

Fill in the Blank

What is the degree of the polynomial f(x) defined below?

f(x) = 7 - 8x³ - 9x⁵ - x + x​⁶

Fill in the Blank

What is the leading coefficient of the polynomial f(x) defined below?

f(x) = 7x - 8x⁵ + 9x⁶ - 10

​The function f(x) is graphed below. What is true about the graph on the interval from point f to point g?

A. It is positive and increasing

B. It is positive and decreasing

C. It is negative and decreasing

D. It is negative and increasing​

​

​The function f(x) is graphed below. What is true about the graph on the interval from point f to point g?

A. It is positive and increasing

B. It is positive and decreasing

C. It is negative and decreasing

D. It is negative and increasing​

​

​Graph (y-axis) below 0 and raise up to right

Answer: D

​

​The function f(x) is graphed below. What is true about the graph on the interval from point a to point b?

A. It is positive and increasing

B. It is positive and decreasing

C. It is negative and decreasing

D. It is negative and increasing​

​

​The function f(x) is graphed below. What is true about the graph on the interval from point a to point b?

A. It is positive and increasing

B. It is positive and decreasing

C. It is negative and decreasing

D. It is negative and increasing​

​

​Graph (y-axis) below 0 and fall down to right

Answer: C

​

​The function f(x) is graphed below. What is true about the graph on the interval from point d to point e?

A. It is positive and increasing

B. It is positive and decreasing

C. It is negative and decreasing

D. It is negative and increasing​

​

​Graph (y-axis) below 0 and fall down to right

Answer: B

​

Multiple Choice

The function f(x) is graphed below. What is true about the graph on the interval from point c to point d?

• It is positive and increasing

• It is positive and decreasing

• It is negative and decreasing

• It is negative and increasing​

Multiple Choice

The function f(x) is graphed below. What is true about the graph on the interval from point d to point e?

• It is positive and increasing

• It is positive and decreasing

• It is negative and decreasing

• It is negative and increasing​

Multiple Choice

The function f(x) is graphed below. What is true about the graph on the interval from point b to point c?

• It is positive and increasing

• It is positive and decreasing

• It is negative and decreasing

• It is negative and increasing​

​Find all possible rational roots of the function

F(x) = 2x³ - 3x² - 12x - 28

​

A. ±1, ±2, ±7, ±14, ±1/7, ±2/7​

B. ±1, ±2, ±7, ±14, 1/2, 7/2

​C. ±1, ±2, ±7, ±14, ±28, ±1/2, ±7​/2

​D. 1, 2, 7, 14, 28, 1/2, 7/2

​Find all possible rational roots of the function

F(x) = 2x³ - 3x² - 12x - 28

​

A. ±1, ±2, ±7, ±14, ±1/7, ±2/7​

B. ±1, ±2, ±7, ±14, 1/2, 7/2

​C. ±1, ±2, ±7, ±14, ±28, ±1/2, ±7​/2

​D. 1, 2, 7, 14, 28, 1/2, 7/2

• P = ±​ (factor of constant = -28) = ± {1, 2, 7, 14, 28}

• Q =±​ (factor of leading coefficient = 2) = ± {1,2}

• P/Q = ​±{1, 2, 7. 14, 28, 1/2, 7/2}

Multiple Choice

Find all possible rational roots of the function

F(x) = 4x³ + x² - x + 11

• ±1, ±2, ±4, ±1/11, ±2/11, ±4/11

• ±1, ±11, ±1/2, ±1/4, ±11/2, ±11/4

• ±1, ±2, ±4

• ±1, ±11

​Find all possible rational roots of the function

F(x) = x³ + x² - x + 15

​A. Factors to: f(x) = (x + 3)(x² - 2x + 5) and Zeros: (-3, 1 + 2i, 1 - 2i)

B. Factors to: f(x) = (x + 3)(x² - 2x + 3) and Zeros: (-3, 1 + i√2, 1 - i√2)

​C. Factors to: f(x) = (x + 3)(x² - 2x + 5) and Zeros: (-3, -2 + i, -2 - i)

​D. Factors to: f(x) = (x + 4)(x² - 2x + 5) and Zeros: (-4, 1 + 2i, 1 - 2i)

​Find all possible rational roots of the function

F(x) = x³ + x² - x + 15

​A. Factors to: f(x) = (x + 3)(x² - 2x + 5) and Zeros: (-3, 1 + 2i, 1 - 2i)

B. Factors to: f(x) = (x + 3)(x² - 2x + 3) and Zeros: (-3, 1 + i√2, 1 - i√2)

​C. Factors to: f(x) = (x + 3)(x² - 2x + 5) and Zeros: (-3, -2 + i, -2 - i)

​D. Factors to: f(x) = (x + 4)(x² - 2x + 5) and Zeros: (-4, 1 + 2i, 1 - 2i)

​By Ti-84 plus CE: (2nd + mode, you may do it twice)

• apps​

• PlySmlt2

• ​POLYNOMIAL ROOT FINDER (select Order "3" and Real for "a + bi"

• graph (input coefficients and constant + change the sign operation)​

Multiple Choice

Find all possible rational roots of the function

F(x) = x³ - 9x² + 28x - 30

• Factors to: f(x) =

  (x - 3)(x² - 6x + 10)

  and Zeros:

  (3, 3 + i, 3 - i)

• Factors to: f(x) =

  (x - 3)(x² - 6x + 10) and Zeros:

  (3, 1 + 3i, 1 - 3i)

• Factors to: f(x) =

  (x - 3)(x - 6x + 10) and Zeros:

  (3, 1 + 3i, 1 - 3i)

• Factors to: f(x) =

  (x + 3)(x - 6x + 10) and Zeros:

  (-3, 3 + i, 3 - i)