Hotmath
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Title:
Hotmath
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Hotmath
Chapter: Polynomial Functions Section: Rational Zeros and The Fundamental Theorem of Algebra
 

Problem: 1

If p / q is a rational number in lowest terms and if it is a zero of

f ( x ) = 10 x 4 + ax 3 + bx 2 + cx + 3, what is true about p and q ? List the possible rational zeros of f .

 

Problem: 3

Obtain the possible rational zeros of the given polynomial, using the Rational –Zero Theorem.

P ( x ) = 8 x 3 + 2 x 2 + 4 x +1

 

Problem: 5

Give all the possible rational zeros of f using the rational zero theorem:

f ( x ) = x 4 + 3 x 2 – 36

 

Problem: 7

Give all the possible rational zeros of f using the rational zero theorem:

f ( x ) = 8 x 4 – 2 x 3 + x + 15

 

Problem: 9

Use the Rational–Zero Theorem to find rational zeros of g if g ( x ) = x 3 – 9 x .

Use an alternate technique to find the rational zeros of g .

 

Problem: 11

Tell whether the given x –value is a zero of the function.

f ( x ) = x 3 – 3 x 2 + 9 x – 27, x = 3 i

 

Problem: 13

Use synthetic division to decide which of the factors 1, –1, 3 and –3 are zeros of the function:

f ( x ) = x 3 + 5 x 2 – 9 x – 45

 

Problem: 15

Use synthetic division to decide which of the factors 1, –1, 3 and –3 are zeros of the function:

f ( x ) = x 4 + 4 x 3 + 4 x 2 – 4 x – 5

 

Problem: 17

Choose a statement that is true about the given quantities:

A. The quantity in column A is greater.

B. The quantity in column B is greater.

C. The two quantities are equal.

D. The relationship cannot be determined from the given information.

 

Problem: 19

Solve the given equation. Identify whether the given equation has any multiple roots.

x 2 – 12 x + 36 = 0

 

Problem: 21

Find all the roots of the following equation by factoring and solving the resulting quadratic equations.

z 4 – 16 = 0

 

Problem: 23

Determine all the real zeros of the function:

f ( x ) = x 3 – 9 x 2 – 40 x + 48

 

Problem: 25

Determine all the real zeros of the function:

f ( x ) = x 3 – 9 x 2 + 8 x + 60

 

Problem: 27

Determine all the zeros of the polynomial function.

f ( x ) = x 4 + 4 x 3 x 2 – 16 x – 12

 

Problem: 29

Determine all the zeros of the polynomial function.

f ( x ) = x 3 x 2 + 81 x – 81

 

Problem: 31

Determine all the zeros of the polynomial function.

f ( x ) = x 4 + 8 x 3 + 16 x 2 + 32 x + 48

 

Problem: 33

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

3, 1, 5

 

Problem: 35

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

–1, –3, –8

 

Problem: 37

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

2 i , –2 i , 3

 

Problem: 39

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

5, 5, 3 + i

 

Problem: 41

One of the zeroes of the function f ( x ) = x 3 – 7 x 2 + 13 x – 6 is at x = 2. Determine the other zeroes.

 

Problem: 43

One of the zeroes of the function f ( x ) = 3 x 3 + 5 x 2 + 4 is at x = –2. Determine the other zeroes.

 

Problem: 45

Given one zero of the polynomial function, determine the other zeros.

f ( x ) = x 3 + 2 x 2 + x + 36, –4

 

Problem: 47

Choose any method to find all the real zeroes of f ( x ) = x 4 – 5 x 2 – 7.