1. write a polynomial function of minimum degree with real coefficients whose zeros include those listed. write the polynomial in standard form. (1 point)
4, -14, and 5 + 8i

a) f(x) = x4 - 362.5x2 + 1450x - 4984
b) f(x) = x4 - 9x3 + 32x2 - 725x + 4984
c) f(x) = x4 - 67x2 + 1450x - 4984
d) f(x) = x4 - 9x3 - 32x2 + 725x - 4984

2. state how many imaginary and real zeros the function has. (1 point)
f(x) = x5 + 7x4 + 2x3 + 14x2 + x + 7

a) 3 imaginary; 2 real
b) 4 imaginary; 1 real
c) 0 imaginary; 5 real
d) 2 imaginary; 3 real

3. write a linear factorization of the function.(1 point)
f(x) = x4 + 16x2

a) f(x) = x2 (4x + i)(4x - i)
b) f(x) = x2 (x + 4i)2
c) f(x) = x2 (x + 4i)(x - 4i)
d) f(x) = x2 (4x + i)2

4. using the given zero, find one other zero of f(x). (1 point)
i is a zero of f(x).= x4 - 2x3 + 38x2 - 2x + 37.

a) -1 + i
b) -i
c) -1 - i
d) 1

5. state the domain of the rational function. (1 point)
f(x) = seven divided by quantity fourteen minus x.

a) all real numbers except 14
b) all real numbers except -14 and 14
c) all real numbers except 7
d) all real numbers except -7 and 7

6. state the vertical asymptote of the rational function.(1 point)
f(x) = quantity x minus seven times quantity x plus four divided by quantity x squared minus four.

a) x = 2, x = -2
b) none
c) x = 7, x = -4
d)x = -7, x = 4

7. state the horizontal asymptote of the rational function. (1 point)
f(x) = quantity x plus nine divided by quantity x squared plus eight x plus five.

a) y = x
b) y = 0
c) y = 9
d) none

8. state the horizontal asymptote of the rational function. (1 point)
f(x) = quantity x squared plus nine x minus nine divided by quantity x minus nine.

a) y = 3
b) none
c) y = 9
d) y = -9

9. give an example of a rational function that has a horizontal asymptote of y = 2/9.

Question 1:

Given the three zeros



The other zero will be the conjugate pair of 


Writing these zeros in factorized form of f(x) gives:
 → Multipy out the first two brackets to get




→ Multiply out further




C
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Question 2

Given 
Collecting two terms together to get


Notice that each group has common factor 
Factorise the common factor gives



The zeros are 
and four of 
The zero  will give an imaginary results

So, f(x) has one real root and four imaginary root

B
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Question 3

 

The zeros are
and 
and 

which gives two answers:  and 

So the factorized form is 


C
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Question 4

If  is one of the zeros of  then the other zero would be to conjugate pair of  which is 

B
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Question 5


The denominator must not equal zero because we can't divide a quantity by zero (undefined result)
So, we need to find the value of  that would make the denominator equal to zero




A
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Question 6


To find vertical asymptote, set the denominator to zero then find 'x'



and 

A
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Question 7


The degree of the denominator is higher than the numerator's, hence the horizontal asymptote is y = 0
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Question 8


The degree of numerator is higher than the degree of the denominator, hence no horizontal asymptote

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Question 9

To have a horizontal asymptote, , both numerator and denominator must be on the same polynomial degree, and the leading degree have coefficient of 2 and 9

Example: