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Ron Paul Curriculum 9th Grade Mathematics

Problem Set #41

Use Cramer’s Rule to solve the following linear systems. If you are comfortable with

your skills in solving determinants, you may use your calculator to solve the

determinants.

1. 2x ‒ y = ‒9

x + 2y = 8

2. x ‒ 6y = 3

3x + 2y = 1

3. x ‒ y + 2z = 0

3x + z = 11

‒x + 2y = 0

4. 2x1 + 3x2 ‒ 5x3 = 1

x1 + x2 ‒ x3 = 2

2x2 + x3 = 8

5. 10

7

2

1

5

1

3

1=+− zyx

10

11

2

3

5

2

3

2=++− zyx

5

9

5

4=+− zyx

Ron Paul Curriculum 9th Grade Mathematics Problem Set #42

Do the following Practice problem sets from the Khan Academy. Learn → Math → Algebra II → Polynomial arithmetic → Add and subtract polynomials → Multiply binomials by polynomials


Problem Set #43

For problems 1- 6, find the axis of symmetry and the vertex of the graph. State

whether the graph of the parabola opens up or down, and if it is in standard form,

whether it is narrower, wider or the same width as the function y = x2. Graph the

function and then check your graph with your calculator or an online graphing

website, such as graphsketch.com.

1. y = x2 + 2x + 1

2. y = ( ) 124

1 2++− x

3. y = ‒4x2 + 8x + 2

4. f(x) = ‒2(x ‒ 1)2 ‒ 5

5. g(x) = 225

3 2 ++− xx

6. y = (x ‒ 3)2

For problems 7 and 8, find the x-intercepts, the axis of symmetry, and the vertex.

Graph the function and then check your graph with your calculator or an online

graphing website, such as graphsketch.com.

7. y = ‒(x ‒ 4)(x + 6)

8. f(x) = 4(x ‒ 7)(x + 2)


Do the following problem sets from the Khan Academy. Learn → Math → Algebra I → Quadratic functions & equations → Solving quadratics by factoring


Problem Set #45

1. Which of the following monomials has the highest degree?

(A) 100

(B) 50x

(C) 10x5

(D) ‒x4y

4

(E) ‒4x3y

2z

2. When in expanded form, which of the following is different from the others?

(A) (4x + 9)2

(B) (‒4x ‒ 9)2

(C) | 4x + 9 |2

(D) ( )22 8116 +x

(E) All of them are the same.

3. The solution set of is

(A) {8, 4}

(B) {8, ‒4}

(C) {‒4, ‒4}

(D) {4, ‒4}

(E) {16, 1}

4. t = ‒9 is a root of the equation t2 + 4t ‒ 45. Which of the following statements is

(are) correct for the equation?

I. t ‒ 9 is a factor of the equation.

II. Division of the equation by t ‒ 9 yields the other factor of the equation.

III. t = ‒5 is another root of the equation.

(A) I only

(B) II and III only

(C) III only

(D) I, II and III

(E) None of the statements are correct.

5. Multiply (4x ‒ 5)(6x ‒ 7).

1238

72

=++ xx

6. Add the expressions

4a2 ‒ 3 + 5a,

6a ‒ 2a2 + 2, and

2a2 ‒ 3a + 8

7. Factor x2 + 7x + 12, or state that it cannot be factored.

8. Simplify (x + 2y)(x ‒ 2y)(x2 + 4y

2):

9. Find the equation for the line passing through points (3, 5) and (‒1, 2).

(A) 2x ‒ 4y = ‒5

(B) 3x + 4y = 11

(C) x + 2y = 13

(D) 3x ‒ 4y = ‒11

(E) 3x ‒ 5y = ‒13

10. What is the domain of the function ?

(A) { x | x ≥ 0 }

(B) { x | x ≤ 1 }

(C) { x | 0 ≤ x ≤ 1 }

(D) { x | x ≥ ‒1 }

(E) { x | x ≤ ‒1 }

51)( ++−== xxfy


Do the following problem sets from the Khan Academy. Learn → Math → Algebra I → Quadratics: Multiplying & factoring → Difference of squares intro → Difference of squares → Factor quadratics by grouping Factor the following quadratic expressions: 1. 2x2 + 5x + 3 2. 4r2 +5r + 1 3. 11z2 + 2z ‒ 9 4. 9d2 ‒ 13d ‒ 10


Do the following Practice problem sets from the Khan Academy. Learn → Math → Algebra I → Quadratic functions & equations → Completing the square (intermediate) → Solve equations by completing the square → Completing the square 1. The height y in feet of a baseball t seconds after it is hit is given by the function:

y = ‒16t2 + 96t + 3

Find the maximum height of the baseball. (Hint: the maximum height of the baseball is the y-coordinate of the vertex of the parabola for the equation above.)


Do the following problem sets from the Khan Academy. Learn → Math → Algebra I → Quadratic functions & equations → Quadratic formula Solve the following quadratic equations using the quadratic formula: 1. 8p2 + 8p + 3 = 0 2. 5x2 + 20x + 21 = 0 3. 8n2 ‒ 4n + 2 = 5n ‒ 11 4. 7x ‒ 3x2 = 85 + 2x2 + 2x 5. 19.25 = ‒8.5(2r ‒ 1.75)2


Do the following Practice problem sets from the Khan Academy. Learn → Math → Algebra I → Quadratic functions & equations → Number of solutions of quadratic equations Write a quadratic equation that has the given roots: 1. 6, ‒9

2. 2, 8

5

3. 34±

4. 4

52 i±−

Solve each equation. Check your answers by using the sum and product of the roots. 5. ‒2x2 ‒ 11x ‒ 12 = 0 6. x2 ‒ 8x = 0

7. 03

1

6

12 =−+ xx

8. Hannah launches a toy rocket and uses a timer to determine that it took 18 seconds to return back to the ground. If the rocket’s height is described by the formula:

2

2

1gttvh i −=

where g = 9.8 m/s2, find the initial velocity vi of Hannah’s rocket.


Problem Set #50

1. The equation of the graph in the figure below is given by y = x2 + 3x + c. What is a

possible value for c?

(A) 3

(B) 2

(C) 1

(D) 0

(E) ‒1

2. Solve for x: (3x + 2)(x ‒ 1) = x2 ‒ 2x ‒ 1

(A) x = ‒1 only

(B) x = ‒1, x = ½

(C) x = ‒½ only

(D) x = ‒½, x = 1

(E) x = ‒1, x = ‒½

3. Which quadratic equation has at least one real solution?

I. x2 + 20x + 100 = 0

II. 5x2 + 2x ‒ 16 = 0

III. 4x2 ‒ 7x + 4 = 0

(A) I only

(B) II only

(C) III only

(D) I and II only

(E) I, II and III

x

y

0 x

y

0

4. The function f is defined by x

)x(f+

=1

1. For what values of x is f (f (x))

undefined?

(A) {0}

(B) {‒1, 0}

(C) {‒½, 0}

(D) {‒1, ‒½}

(E) {‒1, ‒2}

5. Which of the following quadratic equations has 1 ‒ i as a root?

(A) x2 ‒ 4 = 0

(B) x2 ‒ 2x + 1 = 0

(C) x2 + 1 = 0

(D) x2 ‒ 2 = 0

(E) x2 ‒ 2x + 2 = 0

6. A supermarket, rectangular in shape and 200 feet by 300 feet, is to be built on a

city block that contains 81,600 square feet. There will be a uniform strip around the

building for parking. How wide is the strip?

7. The roots of x2 + 2x + 5 = 0 are

(A) 3, 4

(B) ‒1 ± 2i

(C) 3 ± 4i

(D) 6, 3

(E) 2 ± 3i

8. If b2 ‒ 4c = 0, where b and c are real numbers, then the roots of the equation

x2 + bx + c = 0

are:

(A) real and equal

(B) real and unequal

(C) complex and equal

(D) complex and unequal

(E) No solution.

9. Determine the quadratic equation whose roots are x = 2 + 3 and x = 2 ‒ 3 .

(A) x2 + 2x ‒ 3 = 0

(B) x2 ‒ 4x + 1 = 0

(C) x2 ‒ 12 = 0

(D) x2 + 3x + 2 = 0

(E) x2 + 4x ‒ 9 = 0

10. If 12

53)(

+

−=

x

xxf , find f

‒1(x).

(A) 32

5

+

−=

x

xy

(B) 53

12

−

+=

x

xy

(C) x

xy

23

5

−

+=

(D) 5

1

3

2−= xy

(E) 5

1

3

2+−= xy


Do the following problem sets from the Khan Academy. Learn → Math → Algebra I → Exponents and radicals → Multiply & divide powers (integer exponents) → Powers of products & quotients (integer exponents) → Properties of exponents challenge (integer exponents)


Do the following Practice problem sets from the Khan Academy. Learn → Math → Algebra II → Polynomial graphs → Zeros of polynomials (factored form) → Positive & negative intervals of polynomials → Zeros of polynomials (multiplicity) → End behavior of polynomials Describe the end behavior of the graph of the following polynomial functions. Use the notation: f (x) → _?_ as x → +∞ and f (x) → _?_ as x → ‒∞. 1. f (x) = 10x4 2. f (x) = x7 + 3x4 ‒ x2

3. f (x) = 0.2x3 ‒ x + 45 4. f (x) = ‒x6 + 4x3 ‒ 3x 5. f (x) = 5x8 + 8x7

Use synthetic substitution to evaluate the following polynomials: 6. 5x3 ‒ 2x2 ‒ 8x + 16; x = 3 7. x3 + 8x2 ‒ 7x + 35; x = ‒6 8. ‒2x4 + 3x3 ‒ 8x + 13; x = 2 9. ‒7x3 + 11x2 + 4x; x = 3 10. 5x3 ‒ 2x2 + 10x ‒ 15; x = ‒1


Problem Set #53

Factor the following polynomials completely:

1. 14x2 ‒ 21x

2. c3 + 9c

2 + 18c

3. y3 ‒ 64

4. ‒5z3 + 320

5. 3m3 ‒ m

2 + 9m ‒ 3

6. y3 ‒ 7y

2 + 4y ‒ 28

7. 4c3 + 8c

2 ‒ 9c ‒ 18

Find the real-number solutions of the following equations:

8. m3 + 6m

2 ‒ 4m ‒ 24 = 0

9. x6 ‒ 4x

4 ‒ 9x

2 + 36 = 0

10. 4z5 = 84z

3


Divide the following polynomials using long division: 1. (5x4 ‒ 2x3 ‒ 7x2 ‒ 39) ÷ (x2 + 2x ‒ 4) 2. (3x3 + 11x2 + 4x + 1) ÷ (x2 + x) Divide the following polynomials using synthetic division: 3. (2x2 ‒ 7x + 10) ÷ (x ‒ 5) 4. (x4 ‒ 5x3 ‒ 8x2 + 13x ‒ 12) ÷ (x ‒ 6) Given the polynomial f (x) and the factor of f (x), factor f (x) completely. 5. f (x) = x3 ‒ 10x2 + 19x + 30; x ‒ 6 Given the polynomial f (x) and a zero of f (x), find the other zeroes. 6. f (x) = 2x3 ‒ 10x2 ‒ 71x ‒ 9; 9 Do the following problem set from the Khan Academy. Learn → Math → Algebra II → Polynomial division → Divide polynomials by linear expressions → Remainder theorem


Problem Set #55

1. If both x and y are not zero, what is the multiplicative inverse to the expression

1

2

−

−

y

x?

(A) y

x 2

(B) y

x 2− (C) 2x

y (D) 2x

y

− (E) x

2y

2. Solve for x: x2 = 2x + 2.

3. Simplify

4

2

32−

−

−

−

z

xy.

4. Which of the following is a factor of x3 + 1000?

(A) x2 + 100

(B) x2 ‒ 100

(C) x2 ‒ 10x + 100

(D) x2 + 10x + 100

(E) x2 ‒ 20x + 100

5. What is the simplified expression for ( )

( )222

22

2

ba

ba

ba

ba

+

+÷

−

−?

(A) ba

ba

−

+ (B)

ba

ba

+−

(C) 22

22

ba

ba

−

+ (D) 22

22

ba

ba

+

− (E) a ‒ b

6. Which of the following numbers has the largest value:

(A) ( )234 (B)

( )324 (C) 166 (D) ( )252 (E)

24

4

1−

7. What is the remainder when x5 ‒ 10x + 2 is divided by x + 2?

8. Factor ‒x3 + 4x

2 + 4x ‒ 16.

9. If xy6 = ‒192 and xy

3 = 24, find x.

10. Do the following division, putting the answer in the most simplified form:

34

2

3

22

2

2 ++

−−÷

+

−

xx

xx

xx

x=


Problem Set #56

List all of the possible rational zeros for the following functions.

1. f (x) = x4 ‒ 10

2. h(x) = 3x3 ‒ 5x

2 ‒ 11x + 3

3. n(x) = 9x6 ‒ 5x

3 + 27

Find all of the zeros for the following functions.

4. f (x) = x3 + x

2 ‒ 80x ‒ 300

5. t (x) = x4 + 10x

3 + 33x

2 + 38x + 8

6. p (x) = x3 + 3x

2 ‒ 25x + 21

7. g (x) = 6x4 + 22x

3 + 11x

2 ‒ 38x ‒ 40

8. f (x) = x5 ‒ 2x

4 ‒ 12x

3 ‒ 12x

2 ‒ 13x ‒ 10


Find all zeros of the following polynomial functions: 1. f (x) = x4 + 5x3 ‒ 7x2 ‒ 29x + 30 2. g(x) = x4 + x3 + 2x2 + 4x ‒ 8 3. h(x) = 2x4 + 13x3 + 19x2 ‒ 10x ‒ 24 Determine the possible numbers of real zeros and non-real, complex zeros, for the following polynomial functions: 4. g(x) = ‒x3 + 5x2 + 12 5. f (x) = x5 + 7x4 ‒ 4x3 ‒ 3x2 + 9x ‒ 15 6. f (x) = x7 + 4x4 ‒ 10x + 25


For each of the problems below:

1) Use finite differences to determine the degree of the given polynomial function.

2) Write a corresponding system of linear equations for the polynomial function.

3) Use your graphing calculator to solve the linear system to find the correct

coefficients and constants for the polynomial function.

4) Check your answer by checking the inputs and outputs of the polynomial you

found.

1. The first five triangular numbers are shown below. Find the polynomial function

that will generate the triangular numbers.

f (1) = 1 f (3) = 6 f (5) = 15 f (6) = 21

f (2) = 3 f (4) = 10

2. A triangular pyramid is shown below:

The triangular pyramidal numbers are the number of marbles in the pyramid for a base

side of a given length. So the first seven triangular pyramidal numbers are: f (1) = 1;

f (2) = 4; f (3) = 10; f (4) = 20; f (5) = 35; f (6) = 56; f (7) = 84. Find the polynomial

function that will generate the triangular pyramidal numbers.

3.

4.

x 1 2 3 4 5 6

f(x) 11 14 9 -4 -25 -54

x 1 2 3 4 5 6

f(x) 5 14 27 41 53 60


The following problem set is optional.

1. Car tires need to be inflated properly. Overinflation or underinflation can cause

premature treadwear. The data table below shows tire life for different inflation values

for a certain type of tire.

(a) Find the polynomial that best fits the data.

(b) Graph the polynomial you found in part (a) along with the data.

(c) Use your result from part (b) to estimate the optimum tire pressure.

2. A baseball is thrown upward, and its height is measured at 0.5 second intervals

using a strobe light. The resulting data are given in the table below.

(a) Plot the data and decide what degree polynomial you think is appropriate for

fitting the data.

(b) Find a polynomial that best fits the data and graph it along with the data.

(c) Find the times when the ball is 20 feet above the ground.

(d) What is the maximum height the ball reaches?

Pressure (lb/in^2) Tire Life (mi)

26 50,000

28 66,000

31 78000

35 81000

38 74000

42 70000

45 59000

Time (s) Height (ft)

0.0 4.2

0.5 26.1

1.0 40.1

1.5 46.0

2.0 43.9

2.5 33.7

3.0 15.8


Do the following problem sets from the Khan Academy. Learn → Math → Precalculus → Conic Sections → Features of a circle from its standard equation → Graph a circle from its standard equation → Write standard equation of a circle → Features of a circle from its expanded equation → Graph a circle from its expanded equation 1. Write an equation of a circle if the endpoints of a diameter are at (‒7, 11) and (5, ‒10). 2. Write an equation of a circle if the center is at (0.5, 0.7) with a radius of 13.5 units.


Do the following problem sets from the Khan Academy. Learn → Math → Algebra I → Quadratic functions & equations → Parabolas intro → Interpret parabolas in context → Interpret a quadratic graph → Graph parabolas in all forms 1. A parabolic mirror has a focus 6.25 feet above the vertex. The latus rectum is 25 feet long. Find an equation for the parabola formed by the mirror. 2. An automobile headlight contains a parabolic reflector. A special bulb with two filaments is used to produce the high and low beams. The filament placed at the focus produces the high beam and the filament placed off the focus produces the low beam. The equation of the cross section of the reflector is y = x2/12. How far from the vertex should the filament for the high beam be placed?


Do the following Practice problem sets from the Khan Academy. Learn → Math → Precalculus → Conic Sections → Graph & features of ellipses → Ellipse standard equation & graph → Foci of an ellipse from equation → Equation of an ellipse from features 1. An equation of an ellipse is

x2 + 9y2 ‒ 4x + 54y + 49 = 0 Find the coordinates of the center and foci and the lengths of the major and minor axes. Then draw the graph. 2. Write the equation for an ellipse where the endpoints of the major axis are at (10, 2) and (‒8, 2). The foci are at (6, 2) and (‒4, 2). 3. Write the equation for an ellipse for which the endpoints of the major axis are at (2, 12) and (2, ‒4), and the endpoints of the minor axis are at (4, 4) and (0, 4).


Do the following Practice problem sets from the Khan Academy. Learn → Math → Precalculus → Conic Sections → Vertices & direction of a hyperbola → Foci of a hyperbola from equation → Equation of a hyperbola from features Find the standard form of the equation, the coordinates of the vertices and foci, the equations of the asymptotes, and graph the following equations. 1. x2 ‒ 36y2 = 36

2. ( ) ( )

125

1

20

6 22

=−−+ xy

3. 25x2 ‒ 4y2 +100x + 24y ‒ 36 = 0


Problem Set #65

There is no Problem Set for Lesson #65.


Problem Set #66

Install the Conic Sections application on your calculator as necessary, and use it to

graph the following equations. Rearrange them as necessary. You will need to identify

the correct conic section from the equation.

1. x2 ‒ 8x + 8y + 32 = 0

2. 9y2 ‒ 16x

2 = 144

3. x2 + y

2 ‒ 6x + 16y ‒ 152 = 0

4. 25x2 + 64y

2 = 1600


Problem Set #67

Use your graphing calculator to solve the following systems of equations.

Approximate your answers to the nearest one hundredth of a unit.

1. x2 + y

2 = 25

y = 2x2 ‒ 2

2. x2 + y

2 = 4

x2 + y

2 = 16

3. x = y2 ‒ 10y + 25

x2 + y

2 = 36

4. (x ‒ 1)2 + y

2 = 9

x2 + 64y

2 = 64

5. x2 + y

2 = 64

9y2 ‒ 4x

2 = 1


Problem Set #68

Solve the following quadratic-linear systems of equations algebraically.

1. y2 = x

2 + 9

y = 6

2. x2 + y

2 = 36

y = x + 2

3. x = y2 ‒ 10y + 25

12y = x + 32

4. (x ‒ 1)2 + y

2 = 9

y = 7 ‒ x

5. y = 2x2

y = x + 3


Problem Set #69

Solve the following quadratic systems of equations algebraically.

1. y2 = x

2 ‒ 25

x2 ‒ y

2 = 7

2. x2 + y

2 = 64

x2 + 64y

2 = 64

3. y2 = x

2 ‒ 7

x2 + y

2 = 25

4. x2 + 2y

2 = 33

x2 + y

2 ‒ 19 = 2x

5. 3x2 ‒ 20y

2 ‒ 12x + 80y ‒ 96 = 0

3x2 + 20y

2 = 80y + 48


Problem Set #70

1. The system of equations

y = (x ‒ 1)2

y = (x + 1)2

has how many points of intersection?

(A) 0

(B) 1

(C) 2

(D) 3

(E) 4

2. Given the system of equations

x ‒ y = 1

x2 ‒ y

2 = 7

what value of x will be a solution of the system?

(A) ‒4

(B) 4

(C) ‒3

(D) 3

(E) The system has no solutions.

3. For what values of x will (x, y) be a solution of the system of equations:

x2 + y

2 = 25

x + y = 1

(A) x = ‒4 and x = 3

(B) x = ‒4 and x = 5

(C) x = ‒3 and x = 4

(D) x = 1 and x = 5

(E) The system has no solutions.

4. The point (2, 3) satisfies the system of equations

4x2 + y

2 = 25

y2 ‒ x

2 = 5

How many additional points satisfy the system?

(A) 0

(B) 1

(C) 2

(D) 3

(E) 4

5. A rectangular box has volume x3 ‒ 8 cubic inches. If the height of the box is x ‒ 2

inches, what is the area of the base of the box, in square inches? (The volume of the

box equals the area of the base times the height.)

(A) x2 + 4

(B) x2 ‒ 2x ‒ 4

(C) x2 ‒ 2x + 4

(D) x2 + 4x + 4

(E) x2 + 2x + 4

6. (i + 1)(3 ‒ i) + (2i ‒ 1) =

(A) ‒6

(B) 1 + 4i

(C) 2 + 4i

(D) 3 + 4i

(E) 4 + 2i

7. If f (x) = 2x + 1 and g(x) = 3x ‒ 1, then f (g(x)) =

(A) 5x

(B) x ‒ 2

(C) 6x ‒ 1

(D) 6x + 2

(E) 6x2 + x ‒ 1

8. Which of the following is an equation of the line that passes through the points (‒2,

1) and (1, 2) in the xy-plane?

(A) x ‒ 3y = ‒5

(B) x + 3y = 5

(C) x + 3y = 1

(D) x + 3y = ‒5

(E) x ‒ 3y = ‒1

9. Which of the following numbers are irrational?

I. 255.

II. 6−

III. π

(A) II only

(B) III only

(C) I and III only

(D) II and III only

(E) I, II, and III

10. If a < 0 < b < c, then each of the following must be true EXCEPT:

(A) ac < ab

(B) a2 < b

2 < c

2

(C) a3 < b

3 < c

3

(D) ab < b2 < bc

(E) a2b < a

2c


Graph the following linear inequalities. 1. x + y > 5 2. y > 6x ‒ 2 3. y ≥ ‒4x + 3 4. y ≤ | x | 5. y + | x | < 3 6. Tickets for a local show cost $5 for adults and $4 for students. In order to cover expenses, the producers must sell at least $2500 worth of tickets. a) Write an inequality that describes this situation. b) Graph the inequality. c) If 175 adult and 435 student tickets are sold, will the producers cover their expenses?


Do the following problem sets from the Khan Academy. Learn → Math → Algebra I → Inequalities (systems & graphs) → Graphs of inequalities → Two-variable inequalities from their graphs → Systems of inequalities graphs


Problem Set #73

Graph the following systems of nonlinear inequalities.

1. y ≤ 9 ‒ x2

x ≥ 0

y ≥ 0

2. x2 + y

2 ≤ 4

x ‒ y ≥ 0

3. y ≥ x3

y ≤ 2x + 4

x + y ≥ 0

4. x2 + 25y

2 ≤ 100

x2 ‒ y

2 ≥ 36

5. x2 + y

2 ≤ 9

2x + y2 ≥ 1


Problem Set #74

Install the Inequality Graphing application on your calculator. Then solve the systems

of inequalities in Problem Set #71 with your calculator.


Problem Set #75

1. The graph to the right is described

by which of the following inequalities?

(A) 2x ‒ y > 0

(B) 2x ‒ y ≥ 0

(C) 2x + y > 0

(D) 2x + y ≥ 0

(E) x + 2y ≥ 0

2. Identify which quadrants of the

xy-plane contain points in the system:

y < 2x

x > 2

(A) I

(B) I, IV

(C) I, II, III

(D) II, III

(E) II, III, IV

3. Which of the following inequalities

could give the graph shown in the

figure to the right?

(A) 3y < 2x

(B) 2x + 3y > 6

(C) 2x ‒ 3y > 6

(D) 2x + 3y < 6

(E) 2x ‒ 3y < 6

4. Simplify x ‒ [x ‒ x(x ‒ y) + y(x ‒ y)]

(A) (x + y)2

(B) (x ‒ y)2

(C) x2 ‒ y

2

(D) x2 + y

2

(E) 0

5. A system of inequalities is given by:

y ≥ px + 4, p > 0

y ≥ qx + 4, q < 0

In what quadrant(s) is (are) the solutions?

(A) I only

(B) I and II only

(C) II only

(D) III and IV only

(E) I, II and IV only

6. An inequality that passes through the y-intercept

of ‒1 is graphed in the figure to the right. What

is the inequality?

(A) 3y ‒ 2x ‒ 1 ≥ 0

(B) 3x ‒ 2y ‒ 2 ≤ 0

(C) 3x ‒ 2y ‒ 2 ≥ 0

(D) 2x ‒ 3y ‒ 3 ≥ 0

(E) 2x ‒ 3y ‒ 3 ≤ 0

7. The blue shaded region in the figure to the right

represents the intersection of the graphs of x ≤ 0,

y ≤ 0, and which of the following inequalities?

(A) 12 −−≤ xy

(B) 12

1+−≤ xy

(C) 12 −−≥ xy

(D) 12

1−−≥ xy

(E) 12

1−≥ xy

8. The function f is defined for all real numbers x by f(x) = ax2 + bx + c, where a, b,

and c are constants and a is negative. In the xy-plane, the x-coordinate of the vertex of

the parabola y = f(x) is ‒1. If t is a number for which f(t) > f(0), which of the following

must be true?

I. ‒2 < t < 0

II. f(t) < f(‒2)

III. f(t) > f(1)

(A) I only

(B) II only

(C) I and III only

(D) II and III only

(E) I, II and III

9. Which of the following, when added to 4a2 + 9, will result in a perfect square for

all integer values of a?

(A) 0

(B) 3a

(C) 6a

(D) 9a

(E) 12a

10. Which of the following statements is true:

I. When P(x) = x4 ‒ 2x

2 ‒ x + 1 is divided by x ‒ 2, the remainder is 7.

II. The cubic P(x) = (x ‒ 3)(x2 + 2x + 1) has three distinct roots.

III. The polynomial P(x) = x4 + ax ‒ b, where a and b are real numbers, must have

at least one root.

(A) I only

(B) II only

(C) III only

(D) I and III only

(E) None of the statements are true.


Do the following Practice problem sets from the Khan Academy. Learn → Math → Algebra (all content) → Rational expressions, equations & functions → Recognize direct & inverse variation State whether each of the following equations represents a direct, inverse or joint variation, and then find the constant of variation. 1. x/5 = y 2. xy = ‒7 3. x = 4y 4. x = 1/y

5. A = 2

1 bh

6. ba2

1

3

2 −=


Problem Set #77

Identify all of the discontinuities or asymptotes in the following rational functions.

Then use a graphing website such as graphsketch.com to graph them.

1. ( )x

xf3

1=

2. ( )( )22

3

−

−=x

xf

3. ( )127

32 +−

−=

xx

xxf

4. ( )( )24

1

−

−=x

xxf

5. ( )6

362

+

−=x

xxf

6. ( )( )( )( )( )9652

30112

++−−

++=

xxxx

xxxf


Problem Set #78

Find all of the vertical and horizontal asymptotes or discontinuities in the following

functions. Then use a graphing website such as graphsketch.com to graph them.

1. ( )2

62 +

=x

xxf

2. ( )624

182

2

−+

+=

xx

xxf

3. ( )( )( )( )( )213

115

+−

+−=

xx

xxxf

4. ( )( )24

1

−

−=x

xxf

5. ( )6

362

+

−=x

xxf

6. ( )( )( )( )( )9652

30112

++−−

++=

xxxx

xxxf


Solution Set #79

Find all of the x- and y-intercepts and vertical asymptotes for the following functions.

Then use long division or synthetic division to find a polynomial that has the same

end behavior as the rational function. Graph the polynomial and the rational functions

together in a sufficiently large viewing area to verify that the end behavior of the two

functions is the same.

1. ( )3

452

−

++=

x

xxxf

2. ( )13

5

−=x

xxf

3. ( )42

23

−

+=x

xxxf

4. ( )3

63 34

−

+−=

x

xxxf

5. ( )( )2

34

1

22

−

−+−=

x

xxxxf


Problem Set #80

1. Which of the following is a factor of 4 ‒ (x + y)2 ?

(A) ‒(x + y)2

(B) x + y

(C) 2 ‒ x + y

(D) 2 + x + y

(E) 4 + x + y

2. 2v(3v2 ‒ 1) ‒ (6 ‒ 8v

3 + 14v) + 3 =

(A) ‒2v3 + 12v ‒ 3

(B) 14v3 + 12v ‒ 3

(C) 14v3 ‒ 14v ‒ 4

(D) 14v3 ‒ 16v ‒ 3

(E) 14v3 ‒ 16v ‒ 6

3. The radius of the sun is approximately 109 meters, and the radius of an oxygen

atom is approximately 10‒12

meter. The radius of the sun is approximately how

many times the radius of an oxygen atom?

(A) 10‒21

(B) 10‒3

(C) 103

(D) 109

(E) 1021

4. Which of the following are the solutions of the equation 2x(1 ‒ 3x) ‒ 1 + 3x = 0 ?

(A) x = 1/2 and x = 1/3

(B) x = 1/2 and x = ‒1/3

(C) x = ‒1/2 and x = 1/3

(D) x = ‒1/2 and x = ‒1/3

(E) x = 0 and x = 1

5. Which of the following gives all values of x for which | x ‒ 2 | ≤ 5 ?

(A) { x | ‒7 ≤ x ≤ 3 }

(B) { x | ‒5 ≤ x ≤ 5 }

(C) { x | ‒3 ≤ x ≤ 7 }

(D) { x | x < ‒5 }

(E) { x | x < ‒7 or x > 3 }

6. Of the following, which is greatest?

(A) ( )532

(B) ( )532

(C) ( )523

(D) ( )523

(E) ( )235

7. The graph of the line with equation ax + by = 1

is shown to the right. Which of the following

must be true?

(A) a > 0 and b < 0

(B) a > 0 and b > 0

(C) a < 0 and b < 0

(D) a < 0 and b > 0

(E) a = 0 and b > 0

8. The set of all values of b for which the equation 4x2 + bx + 1 = 0 has either one real

root or two real roots is defined by

(A) b > 4

(B) b < 4

(C) b ≥ 1 or b ≤ ‒1

(D) b > 4 or b < ‒4

(E) b ≥ 4 or b ≤ ‒4

9. Which quadrants of the xy-plane contain points of the graph 2x ‒ y > 4 ?

(A) I, II, and III only

(B) I, II, and IV only

(C) I, III, and IV only

(D) II, III, and IV only

(E) I, II, III and IV

10. When i

i

+

+

2

43 is expressed in the form a + bi, what is the value of a ?

O x

y

O x

y

ronpaulcurriculum.com · 2020. 8. 29. · given the polynomial f ( x) and a zero of f ( x), find...

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