Free Printable Pre-Assessment 3.1The Rational Root Theorem

1.

According to the Rational Root Theorem, what are the all possible rational roots?
2x³ - 11x² + 12x + 9 = 0

a)

±1,±2

b)

±1,±3,±9

c)

±1,±3±9,±1/2,±3/2,±9/2

d)

±1,±2,±1/3,±2/3,±1/9,±2/9

2.

Use the Complex Conjugate Root Theorem to state another complex solution. Given that 3+5i is a root.

a)

-3+5i

b)

-3-5i

c)

-5i

d)

3-5i

3.

Use the Complex Conjugate Root Theorem to state another complex solution. Given that -7i is a root.

a)

7i

b)

-7i+1

c)

1-7i

d)

1+7i

4.

All imaginary roots comes in pairs.

a)

True

b)

False

5.

What is the conjugate of -√2 + 7

a)

-√2 - 7

b)

√2 - 7

c)

-7 + √2

d)

7 + √2

6.

If a+bi is a root of a polynomial, then _______ is also a root.

a)

i²

b)

a-bi

c)

a+bi

d)

-i²

7.

Solve by using Rational Root Theorem:
x³ + 3x² - 6x - 8 = 0

a)

x = {-4, -1, 2}

b)

x = {-4, -1, ±√2}

c)

x = {1, 2, 4}

d)

x = {4,-1,√2}

8.

Which of the following would be the correct factored form with roots 2, 5i, and 6?

a)

(x-6)(x-2)(x+5i)(x-5i) = 0

b)

(x+6)(x+2)(x+5i) = 0

c)

(x-6)(x-2)(x-5i) = 0

d)

(x+6)(x+2)(x-5i) = 0

9.

Find a polynomial function of degree 3 with zeros 3i and -2.

a)

P(x) = x³+2x²+9x+11

b)

P(x) = x³-2x²+9x+18

c)

P(x) = x³+2x²+9x+18

d)

P(x) = x²-3ix+2x-6i

10.

Write a polynomial function of least degree with integral coefficients that has the given zeros.

-1, 5, -3+2i

a)

P(x) = x⁴+2x³-16x²-82x-69

b)

P(x) = x⁴+7x³-16x²-82x-65

c)

P(x) = x⁴+2x³-16x²-82x-65

d)

P(x) = x⁴+2x³-16x²-82x-68

11.

Write a polynomial function of least degree with integral coefficients that has the given zeros.

5, -2i

a)

P(x) = x³-5x²+4x-20

b)

P(x) = x³-5x²+4x-23

c)

P(x) = x³-5x²+4x-19

d)

P(x) = x³-10x²+4x-20

12.

Find ALL zeros

a)

A

b)

B

c)

C

d)

D

13.

Find all zeros of the polynomial

a)

A

b)

B

c)

C

d)

D

14.

Find all the zeros of this polynomial.

a)

A

b)

B

c)

C

d)

D

15.

Jose Altuve hit a baseball. The path of the ball was modeled by this equation: $h\left(t\right)=-16t^2+96t+3$ where $h\left(t\right)$ is the height of the ball in feet and $t$ is the number of seconds the ball was in the air.

What was the maximum height that the ball reached?

a)

96 feet

b)

147 feet

c)

16 feet

d)

3 feet

16.

Jose Altuve hit a baseball. The path of the ball was modeled by this equation: $h\left(t\right)=-16t^2+96t+3$ where $h\left(t\right)$ is the height of the ball in feet and $t$ is the number of seconds the ball was in the air.

After how many seconds did the ball reach it's maximum height?

a)

96 seconds

b)

147 seconds

c)

16 seconds

d)

3 seconds

17.

Jose Altuve hit a baseball. The path of the ball was modeled by this equation: $h\left(t\right)=-16t^2+96t+3$ where $h\left(t\right)$ is the height of the ball in feet and $t$ is the number of seconds the ball was in the air.

After how many seconds did the ball first reach 131 feet above the ground?

a)

0

b)

1

c)

2

d)

3

18.

Solve the following equation.

x^4-12x^2+11=0

a)

$x\ =\ 1\ and\ -1$

b)

$x\ =\ -\sqrt{11\ }and\ \sqrt{11}$

c)

$x\ =\ -1,\ 1,\ \sqrt{11}\ and\ -\sqrt{11}$

d)

$x\ =\ 1\ and\ \sqrt{11}$

19.

Factor and solve.

3n^3-4n^2+9n=12

a)

$\frac{4}{3}$

b)

$\pm i\sqrt{3}$

c)

$\frac{4}{3},\ \pm i\sqrt{3}$

d)

$\frac{3}{4}$

20.

Find all zeros for the polynomial function P(x) = x³ -3x²-5x+15

a)

3, √5

b)

3, √5, -√5

c)

-3, √5, -√5

d)

-3, 5i, -5i

21.

Find all zeros of the polynomial function P(x) = x³+6x²+9x+54

a)

6, 3, -3

b)

-6, 3, -3

c)

-6, 3i, -3i

d)

6, 3i, -3i

22.

Find all zeros for the polynomial function P(x)= x³-3x²+12x-10

a)

1, 1-3i, 1+3i

b)

1, 1+3i, -1-3i

c)

-1, -1+3i, -1-3i

d)

-1, -5, 2

23.

Which of the following is a COMPLETE list of all possible Rational Zeros?
f(x) = 3x³ + 2x² - 6x + 7

a)

±1, ±7, ±1/3, ±7/3

b)

±1, 7

c)

±1, ±3, ±7, ±7/3

d)

±1, ±3, ±7, ±1/3, ±7/3

24.

There are 3 potential options for the number of real zeros for this polynomial function.
$f(x)=2x^5-3x^2+4x-10$

Select the 3 that apply.

a)

5

b)

4

c)

3

d)

2

e)

1

25.

What are the solutions to the polynomial: p(x)=(2x-1)(x² +3x +1)
(Hint: Use quadratic formula for x²+3x+1)

a)

x=1/2, (-3+√5)/2, (-3-√5)/2

b)

x=1/2, (3+√5)/2, (3-√5)/2

c)

x=-1/2, (-3+√5)/2, (-3-√5)/2

d)

x=-1/2, (3+√5)/2, (3-√5)/2

26.

Which of the following statements is false for the given function? $g\left(x\right)=2x^2-3x-2$

a)

$\left(x-2\right)\$ is a factor

b)

$f\left(2\right)=0$

c)

$x=2$ is a solution

d)

$x=1$ is a solution

e)

$x=-1$ is a solution

27.

Which of the following statements is true for the given polynomial function? $p\left(x\right)=2x^4+x^3-6x^2-7x-2$

a)

$f\left(1\right)=0$

b)

$f\left(-2\right)=0$

c)

$f\left(0\right)=-1$

d)

$f\left(-1\right)=0$

28.

Which of the following is a COMPLETE list of all possible Rational Zeros?
f(x) = 3x³ + 2x² - 6x + 7

a)

±1, ±7, ±1/3, ±7/3

b)

±1, 7

c)

±1, ±3, ±7, ±7/3

d)

±1, ±3, ±7, ±1/3, ±7/3

29.

According to the Rational Root Theorem, what are the all possible rational roots?
2x³ - 11x² + 12x + 9 = 0

a)

±1,±2

b)

±1,±3,±9

c)

±1,±3±9,±1/2,±3/2,±9/2

d)

±1,±2,±1/3,±2/3,±1/9,±2/9

30.

Solve by using Rational Root Theorem:
x⁴ - x³ + 2x² - 4x - 8 = 0

a)

-1, 2, ±2

b)

-1, 2, ±2i

c)

1, -2, 4

d)

1, 2, 2i

31.

How many zeros does the following function have?
f(x)= x⁵- 3x³ + x

a)

8

b)

9

c)

5

d)

3

32.

$3x^7-5x^4+7x^2-8=0$

Identify the P and Q values in the equation:

a)

P = -5 ; Q = 3

b)

P = 8; Q = 3

c)

P = -8; Q = 7

d)

P = 3; Q = 8

33.

$2x^4-17x^3+39x^2-35x+11=0$ Find the roots.

a)

{11/2 , 1 mult. 3}

b)

{-1mult. 3, 3/2}

c)

{1, -2, 3/2, 4}

d)

{-1, -2, 5/2, -4}

34.

$4x^4-4x^3-15x^2-2x+5=0$ Find the roots.

a)

{1 mult.2, 5/2, 1/2}

b)

{-1 mult. 2, -1/2, 3/2}

c)

{1 mult. 2, -3/2, -1/2}

d)

{-1 mult. 2, 5/2, 1/2}

35.

$6x^4-11x^3-21x^2-x+3=0$ Find the roots.

a)

{-1, -3, 1/2, 1/3}

b)

{1/3, -1/2, -1, 3}

c)

{1, -1, -3, 1/2}

d)

{-1, -3, 2/3, -1/2}

36.

$4x^4-25x^2+36=0$
Find the roots.

a)

$\pm\frac{3}{2},\ \pm2$

b)

$\pm2,\ \pm\frac{2}{3}$

c)

$1,\ -2,\ -1,\ 3$

d)

$2,\ -3,\ \frac{2}{3},\ -2$

37.

True or False

When a polynomial equation has the following factors, (x+3)(x+3)(x-2), we say that x = -3 has a multiplicity of 3.

a)

True

b)

False

38.

Solve the following equation. x⁴+4x³ + 3x²= 0

a)

x = 0, x = 3, x = 1

b)

x = -3, x = -1

c)

x = 3, x = 1

d)

x = 0, x = -3, x = -1

39.

Solve by using Rational Root Theorem:
x³ - 7x - 6 = 0

a)

1, 2, 3

b)

-1, -2, - 3

c)

-1, -2, 3

d)

1,-2,-3

40.

Write a polynomial function f of least degree that has rational coefficients, a leading coefficient of 1, and the given zeros.

1, -6, 4

a)

x³+x²-26x+24

b)

x³+9x²-14x+24

c)

x³-x²+14x+24

d)

x³-x²+26x+24

41.

What are the values of x when (2x+5)(x-6)=0?

a)

-6, 5/2

b)

-5/2, 6

c)

6, 2

d)

5, -6

42.

Below are four possible zeros for this polynomial.
Figure out which one "works" and can be used to find the others.
f(x) = x³ - 7x² + 2x + 40

a)

8

b)

2

c)

5

d)

40

43.

Q. Is (x-2) a factor of f(x) = x³- 8x²+ 14x - 4?

a)

Yes, (x-2) is a factor. There is a remainder.

b)

Yes, (x-2) is a factor. The remainder is zero.

c)

No, (x-2) is not a factor. The remainder is zero.

d)

No, (x-2) is not a factor. There is a remainder.

44.

Below are four possible zeros for this polynomial.
Figure out which one "works" and can be used to find the others.
f(x) = x³ - 8x² - 23x + 30

a)

3

b)

-1

c)

-10

d)

1

45.

Which of these is a possible rational root of the polynomials?

x³ - 6x² - x + 30

a)

0

b)

4

c)

-9

d)

-15

46.

What is the degree of the following polynomial that has 4 real roots and 2 complex roots?

a)

4

b)

2

c)

8

d)

6

47.

All imaginary roots comes in pairs.

a)

True

b)

False

48.

What is the degree of a polynomial that has roots of -2, 4i and -4i ?

a)

3

b)

4

c)

1

d)

2

49.

If 2i is a zero, then

a)

2i repeats twice

b)

-2i is a zero too

c)

1-2i is a zero too

d)

0+2i is a zero too

50.

The degree of the polynomial determines the number of roots.

a)

True

b)

False

51.

Which formula is the Fundamental Theorem of Algebra Formula?

a)

There are infinitely many rationals between two reals.

b)

Every polynomial equation having complex coefficents and degree greater than the number 1 has at least one complex root.

c)

There are infinitely many prime numbers.

d)

All numbers are rational numbers.

Pre-Assessment 3.1The Rational Root Theorem

Total questions: 51

Worksheet time: 3hrs 8mins

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