Module 02: Polynomial and Rational Functions

Question 1

Find the vertex of the graph of the function.

f(x) = (x + 4)² + 4 (2 points)

1. (0, -4)
2. (4, 0)
3. (4, -4)
4. (-4, 4)

Answer 1

4. (-4, 4)

Question 2

Find the vertex of the graph of the function.

f(x) = 2x² + 8x + 10 (2 points)

1. (3, -1)
2. (2, -2)
3. (-2, 2)
4. (-1, 3)

Answer 2

3. (-2, 2)

Question 3

Write the quadratic function in vertex form.

y = x² + 8x + 18

Answer 3

y = (x + 4)² + 2

Question 4

A projectile is thrown upward so that its distance above the ground after t seconds is given by the function h(t) = -16t2 + 704t. After how many seconds does the projectile take to reach its maximum height? Show your work for full credit. (2 points)

Answer 4

The projectile will take 22 seconds (t) to reach its maximum height at 7744.

Question 5

Find the zeros of the function.

f(x) = 9x² - 27x + 20

1. 5 and 4
2. -5 and -4
3. -5/3 and -4/3
4. 5/3 and 4/3

Answer 5

4. 5/3 and 4/3

Question 6

Find the zeros of the function.

f(x) = 9x³ - 45x² + 36x (2 points)

1. 0, 1, and 4
2. -1 and -4
3. 1 and 4
4. 0, -1, and -4

Answer 6

1. 0, 1, and 4

Question 7

Find the zeros of the polynomial function and state the multiplicity of each.

f(x) = 4(x + 7)²(x - 7)³

1. 4, multiplicity 1; -7, multiplicity 3; 7, multiplicity 3
2. -7, multiplicity 3; 7, multiplicity 2
3. 4, multiplicity 1; 7, multiplicity 1; -7, multiplicity 1
4. -7, multiplicity 2; 7, multiplicity 3

Answer 7

4. -7, multiplicity 2; 7, multiplicity 3

Question 8

Find a cubic function with the given zeros. (2 points)

√2 , -√2 , -2

1. f(x) = x³ + 2x² - 2x + 4
2. f(x) = x³ +2x²+ 2x - 4
3. f(x) = x³ - 2x² - 2x - 4
4. f(x) = x³ + 2x² - 2x - 4

Answer 8

4. f(x) = x³ + 2x² - 2x - 4

Question 9

Expand the following using either the Binomial Theorem or Pascal’s Triangle. You must show your work for credit.

(x - 5)⁵

Answer 9

See attachment

Question 10

Divide using synthetic division, and write a summary statement in fraction form. (2 points)

Answer 10

Answer: 3

Question 11

Divide f(x) by d(x), and write a summary statement in the form indicated.

f(x) = x⁴ + 4x³ + 6x² + 4x + 5; d(x) = x² + 1

1. f(x) = (x² + 1)( x² + 4x + 5) + 12x - 15
2. f(x) = (x² + 1)( x² + 4x + 5)
3. f(x) = (x² + 1)( x² - 4x + 5)
4. f(x) = (x² + 1)( x² - 4x + 5) + 12x - 15

Answer 11

2. f(x) = (x² + 1)( x² + 4x + 5)

Question 12

Find the remainder when f(x) is divided by (x - k).

f(x) = 3x³ - 4x² - 3x + 14; k= 3

• 50
• 68
• -12
• 112

Answer 12

50

Question 13

Use synthetic division to determine whether the number k is an upper or lower bound (as specified) for the real zeros of the function f.

k = -1; f(x) = 4x³ - 2x² + 2x + 4; Lower bound?

YES or NO

Answer 13

YES

Question 14

Use the Rational Zeros Theorem to write a list of all possible rational zeros of the function.

f(x) = 2x³ + 8x² + 7x - 8

Answer 14

According to the Rational Zeros Theorem, p is the factors of the constant term of a0, which is 8. Therefore, p = 1, 2, 4, 8. Next, q is the leading coefficient of an, which is 2, so the factors of q are 1 and 2. According to the Rational Zeros Theorem, the possible zeros is p/q.

Question 15

Write the sum or difference in the standard form a + bi. (2 points)

( 7 + 5i) - ( -9 + i)

1. 16 + 4i
2. -16 - 4i
3. 16 - 4i
4. -2 + 6i

Answer 15

1. 16 + 4i

Question 16

Write the product in standard form. (2 points)

( 4 + 7i)( 7 + 7i)

Answer 16

-21 + 77i

Question 17

Find the product of the complex number and its conjugate

1 + 3i

1. 1 + 9i
2. 10
3. -8
4. 1 - 9i

Answer 17

2. 10

Question 18

Write the expression in standard form:

3/ (3-12i)

Answer 18

1/17 + 4/17i

Question 19

Find the real numbers x and y that make the equation true

-4 + yi = x + 3i

Answer 19

According to the Rule of Equality of Complex Numbers, a+bi=c+di, only if a=c and b=d. Therefore, in order for the above equation to be equal, x=-4 and y=3.

Question 20

Write a polynomial function of minimum degree with real coefficients whose zeros include those listed. Write the polynomial in standard form. (2 points)

4, -8, and 2 + 5i

Answer 20

f(x) = x⁴ - 19x² + 244x - 928

Question 21

State how many imaginary and real zeros the function has. (3 points)

f(x) = x³ + 5x² + x + 5

1. 0 imaginary; 3 real
2. 1 imaginary; 2 real
3. 3 imaginary; 0 real
4. 2 imaginary; 1 real

Answer 21

4. 2 imaginary; 1 real

Question 22

Write a linear factorization of the function.

f(x) = x⁴ + 64x²

Answer 22

f(x) = x²(x + 8i)(x - 8i)

Question 23

State the domain of the rational function. (2 points)
f(x) = 13/ (10-x)

1. All real numbers except -10 and 10
2. All real numbers except 13
3. All real numbers except 10
4. All real numbers except -13 and 13

Answer 23

3. All real numbers except 10

Question 24

State the vertical asymptote of the rational function. (2 points)

f(x) = [(x-6)(x+6)] / (x²-9)

Answer 24

x = 3, x = -3

Question 25

State the horizontal asymptote of the rational function. (2 points) **f(x) = (5x+1) / (9x-2)** 1. None 2. y = 3/2 3. y = 5/9 4. y = 0

Answer 25

**3.** y = 5/9

Question 26

State the horizontal asymptote of the rational function. (2 points) **f(x) = (x+9) / (x²+2x+3)**

Answer 26

y = 0

Question 27

Find the vertex of the graph of the function. (5 points) **f(x) = (x + 6)² + 3** 1. (0, -6) 2. ( 3, 0) 3. ( 3, -6) 4. ( -6, 3)

Answer 27

**4.** ( -6, 3)

Question 28

Find the vertex of the graph of the function. (5 points) ## Footnote **f(x) = 2x² - 8x + 6**

Answer 28

(2, -2)

Question 29

Find the axis of symmetry of the graph of the function. (5 points) **f(x) = 2x² - 4x** 1. x = 1 2. x = 3 3. x = 0 4. x = -2

Answer 29

1. x = 1

Question 30

Write the quadratic function in vertex form. (5 points) **y = x² - 2x + 5** 1. y = (x + 1)² - 4 2. y = (x - 1)² - 4 3. y = (x + 1)² + 4 4. y = (x - 1)² + 4

Answer 30

**4.** y = (x - 1)² + 4

Question 31

A projectile is thrown upward so that its distance above the ground after t seconds is h = -16t² + 440t. After how many seconds does it reach its maximum height? (5 points)

Answer 31

14 seconds

Question 32

. Find the zeros of the function. (5 points) f(x) = x² - 6x + 8

Answer 32

4 and 2

Question 33

Find the zeros of the function. ## Footnote **f(x) = 4x³ - 12x² - 40x**

Answer 33

0, 5, -2

Question 34

Find the zeros of the polynomial function and state the multiplicity of each. (5 points) **f(x) = 5(x + 6)²(x - 6)³**

Answer 34

-6, multiplicity 2; 6, multiplicity 3

Question 35

Find a cubic function with the given zeros. (5 points) **-2, 5, -6**

Answer 35

f(x) = x³ + 3x² - 28x - 60

Question 36

Divide using synthetic division, and write a summary statement in fraction form.

Answer 36

Answer: **2**

Question 37

Use the Rational Zeros Theorem to write a list of all potential rational zeros. ## Footnote **f(x) = 14x³ + 56x² + 2x - 7**

Answer 37

±1, ± 1/2, ±7, ± 7/2, ± 1/7, ± 1/14

Question 38

Write the expression in standard form: 5/ (2-14i)

Answer 38

1/20 + 7/20 i

Question 39

Write a polynomial function of minimum degree with real coefficients whose zeros include those listed. Write the polynomial in standard form. (5 points) ## Footnote **4, -8, and 2 + 3i**

Answer 39

f(x) = x⁴ - 35x² + 180x - 416

Question 40

State how many imaginary and real zeros the function has. (5 points) **f(x) = x⁴ - 15x² - 16** 1. 4 imaginary; 0 real 2. 3 imaginary; 1 real 3. 2 imaginary; 2 real 4. 0 imaginary; 4 real

Answer 40

**3.** 2 imaginary; 2 real

Question 41

Write a linear factorization of the function. (5 points) **f(x) = x⁴ + 81x²**

Answer 41

f(x) = x² (x + 9i)(x - 9i)

Question 42

Using the given zero, find one other zero of f(x). (5 points) 3 - 6i is a zero of *f(x).= x⁴ - 6x³ + 46x² - 6x + 45.* 1. **3 - i** 2. **1** 3. **3 + 6i** 4. **3 + i**

Answer 42

**3.** 3 + 6i

Question 43

State the domain of the rational function. (5 points) f(x) = 17/(5-x)

Answer 43

All real numbers except 5.

Question 44

What is the quadratic function?

Answer 44

***f(x) = ax² + bx + c*** a, b, c → real numbers (not zero) * u-shaped \> parabola * symmetric **Vertex**: interception with axis Opens upwards: leading coefficient is positive Opens downwards: leading coefficient negative * if a \> 0, then the parabola opens upward, and if a \< 0, the parabola opens downward.* * * * Simplest: **f(x) = ax²** Find the vertex: **-b/2a** *a² + bx + c*

Question 45

How do you find the vertex from the quadratic function?

Answer 45

Find the vertex: **-b/2a**

Question 46

What is the vertex form?

Answer 46

***f(x) = a(x - h)² + k*** * Vertex point: (h, k) * axis x = h

Question 47

What is the quadratic formula?

Answer 47

Question 48

What is the Binomial Theorem?

Answer 48

Question 49

What is Pascal's Triangle?

Answer 49

Question 50

What is the Leading Coefficient Test?

Answer 50

Rise and falling of the graph dependent: 1. **degree is odd or even** * if the degree is even, the graph will rise at both ends, * if the degree is odd, then the graph will rise at one end and fall at the other end 2. **leading coefficient test** * coefficient positive: falls to the left and rises to the right * coefficient negative: rises to the left and falls to the right

Question 51

What is the *Zeros of Polynomial Functions* test?

Answer 51

Zeros → interception with x-axis 1. ***x = a*** is a zero of the function 2. ***x = a*** is a solution for the polynomial set equal to 0 3. ***(x - a*)** is a factor of the polynomial 4. **(*a*, 0)** is an x-intercept of the function

Question 52

What is the Intermediate Value Theorem?

Answer 52

Let *a* and *b* real numbers such that **a \< b.** If f(a) ≠ f(b), then interval (a, b), f takes on every value between f(a) and f(b) * if (a, f(a)) and (b, f(b)) are 2 points → every x value between a and b takes on every y-value between f(a) and f(b)

Question 53

What is the Extreme Value Theorem?

Answer 53

if function f(x) continuous on a closed interval [a, b] then f(x) has both a maximum and a minimum on [a, b] theorem is stating that if you graph a curve from x = a and never pick your pencil up until x = b, then that curve is guaranteed to have a maximum and minimum: 1. x values at the endpoint of the curve [$x = a$ & $x = b$] 2. any x value where the curve hits a peak or a valley

Question 54

What is the Remainder Theorem?

Answer 54

If a polynomial f(x) ÷ (x - k), the remainder is r = f(k). ## Footnote *This means we can find a point on the graph of a function by choosing a value k for x = k and using division to get a remainder. The remainder will be the y coordinate of the point on the graph.*

Question 55

What is the Factor Theorem?

Answer 55

A polynomial f(x) has a factor ( - k) if, and only if, f(k) = 0. If remainder is a 0 = being a factor

Question 56

What is the Rational Zero Test?

Answer 56

If: it has a integer coefficient ***f(x) = a_nxⁿ = a_n-1x^n-1 + ... + a₁x + a₀ *** Every rational zero of f(x) has a form: Rational Zero * **p/q** (p and q have no common factors other than 1) * ***p*** factor of the constant term a0 * ***q*** factor of the leading coefficient of an *Note that the Rational Zero Test does not guarantee that any of these values will be zeros; it just states that if the zeros are rational, then they will come from this list.*

Question 57

What is the Descartes' Rule of Signs?

Answer 57

1. Number of positive zeros: * equal to the number of sign changes between consecutive coefficients in f(x) * less than said number by a multiple of 2 2. Number of negative zeros: * equal to the number of sign changes between f(-x) * less than said number by a multiple of 2

Question 58

What is the steps to find the zeros?

Answer 58

_Step 01_: Use the Rational Zero test to determine the list of possible zeros _Step 02_: Use the Descartes' Rule of Signs to narrow it down _Step 03_: Use sythetic division to see which of the remainders is an actual zero

Question 59

What is the Bounds for Real Zeros?

Answer 59

f(x) is divided by (x-c) then: 1. **c \> 0** → each number in the last row is positive or zero * c upper bounds for real zeros of f * upper bound = positive 2. **c \< 0** → number in the last row are alternately positive and negative * c lower bounds for real zeros of f * lower bounds = negative

Question 60

What is the Rule of the Equality of Complex Numbers?

Answer 60

Question 61

What is the Rule of the Addition and Subtraction of Complex Numbers?

Answer 61

if a + bi and c + di are two complex numbers: **Sum**: *(a + bi) + (c + di) = (a + c) + (b + d)i* **Difference**: *(a + bi) - (c + di) = (a - c) + (b - d)i* Some of the properties for real numbers are also valid for complex numbers. They include the _associative property of addition and multiplication_, the _commutative property of addition and_ _multiplication_, and the _distributive property of multiplication over addition_.

Question 62

How do you plot complex numbers?

Answer 62

x-axis → real axis y-axis → imaginary axis *a + bi* * a = real axis * b = imaginary axis

Question 63

What are complex conjugates?

Answer 63

Each of two complex numbers having their real parts identical and their imaginary parts of equal magnitude but opposite sign. z = a + bi z = a - bi

Question 64

How does division with complex numbers work?

Answer 64

_Step 01:_ Determine complex conjugate of the denominator _Step 02:_ Multiple the numerator and the denominator with the complex conjugate

Question 65

What is the Fundamental Theorem of Algebra?

Answer 65

**If f(x) is a polynomial of degree n, where n \> 0, f has at least one zero in the complex number system.** expand our set of zeros to the complex numbers, we can say that every nth degree polynomial has exactly n zeros. set of complex numbers → one imaginary number and one real number

Question 66

What is the Linear Factorization Theorem?

Answer 66

If f(x) has a polynomial degree of n: ***f(x) = a_nxⁿ + a_n-1x^n-1 + ... + a₁x + a₀ *** *n \> 0*, *f* has precisely *n* linear factors ***f(x) = a_n(x-c₁)(x-c₂)(x-c₃) ... (x-c_n)*** * c1, c2, c3 → complex numbers * an → leading coefficient

Question 67

What are Rational Functions?

Answer 67

Question 68

What is the vertical and horizontal asymptotes?

Answer 68

Horizontal Asymptotes The line *x = a* is a vertical asymptotes of the graph f is f(x) approaches b as x approaches infinity or x approaches negative infinity Vertical Asymptotes The line *y = b* is a horizontal asymptote of the graph f is f(x) approaches b as x approaches infinity or x approaches negative infinity

Question 69

What are the asymptotes of rational functions?

Answer 69

Question 70

What are the 5 characteristics of graphs of rational functions?

Answer 70

1. the y-intercept (if any) is the value of f(0) 2. the x-intercept (if any) are the zeros of the numerator, the solution is p(x) = 0 3. **vertical** asymptotes (if any) are the zeros of the denominator, the solution is p(x) = 0 4. **horizontal** asymptotes (if any) is the value that f(x) approaches as x increases or decreases 5. determine the behavior of the graph between and to the left and right of each x-intercept

Question 71

What are slant asymptotes?

Answer 71

numerator has exactly one more degree than the denominator → slant asymptote * **Example** * For the function f(x) = (x2 - 4x + 3) / (x + 2), find the slant asymptote.* * Dividing the denominator into the numerator gives (x - 6) + 15 / (x + 2), and the slant asymptote would be y = x - 6.*

Question 72

What is the difference between the vertical asymptotes and holes?

Answer 72

Vertical asymptotes: _the zeros of q(x)_ *R(x) = p(x)/q(x)* **Holes in the Asymptotes:** Zeros of the denominator gets cancelled out when R(x) is put in a lowest term