Homework 3.3

1.  Decide whether the following statement is true or false. If​ false, tell why.

Since x−1 is a factor of f(x)=

−

+

−2​, we can conclude that f(1)=0.

Is the statement true or​ false?

1. The statement is true.
2. The statement is false because since x−1 is a factor of f(x)=
   $x^6–x^4+2x^2–2$

   ​, we can conclude that f(0)=1.

3. The statement is false because since x−1 is a factor of f(x)=
   $x^6–x^4+2x^2–2$

   ​, we can conclude that f(−1)=0.

2. Determine whether the following statement is true or false. If​ false, explain why.

Because f(2)=0 for f(x)=

​, we can conclude that x−2 is a factor of​ f(x).

Is the statement true or​ false?

1. The statement is false. Because f(2)=0 for f(x)=
   $x^6–x^4+6x^2–72$

   ​, we can conclude that2 is a factor of​ f(x), not x−2.

2. The statement is true.
3. The statement is false. Because f(2)=0 for f(x)=
   $x^6–x^4+6x^2–72$

   x6−x4+6×2−72​, we can conclude that x is a factor of​ f(x), not x−2.

4. The statement is false. Because f(2)=0 for f(x)=
   $x^6–x^4+6x^2–72$

   x6−x4+6×2−72​, we can conclude that x+2 is a factor of​ f(x), not x−2.

3.  Decide whether the following statement is true or false. If​ false, tell why.

For  

, 5 is a zero of multiplicity 5.

Choose the correct answer below.

1. The statement is true.
2. The statement is false because 5 is not a zero of f(x).
3. The statement is false because 5 is a zero of multiplicity ____

4.  Use the factor theorem and synthetic division to decide whether the second polynomial is a factor of the first.

$x^3+8x^2+4x-48$

; x + 6

Is x+6 a factor of

1. Yes
2. No

5. Use the factor theorem and synthetic division to decide whether the second polynomial is a factor of the first.

$2x^4+2x^3-3x^2+5x+2$

​; x+2

Is x+2 a factor of

1. Yes
2. No

6. Use the factor theorem and synthetic division to decide whether the second polynomial is a factor of the first.

$-x^3+7x-6$

​; x+3

Is x+3 a factor of

?

1. Yes
2. No

7.  Use the factor theorem and synthetic division to decide whether the second polynomial is a factor of the first.

$x^3+8x^2+9$

; x−3

Is x−3 a factor of 

1.  ​Yes, because ​f(3​)=0.
2. ​No, because ​f(3​)= ____

8.   Use the factor theorem and synthetic division to decide whether the second polynomial is a factor of the first.

$3x^4+4x^3-39x^2+84x+20$

​; x+5

Is x+5 a factor of

1. Yes
2. No

9. Factor​ f(x) into linear factors given that k is a zero of​ f(x).

f(x)=

​; k= 1

f(x)

10. Factor ​f(x)= 

into linear factors given that −8 is a zero of​ f(x).

11. Factor ​f(x) = 

into linear factors given that −5 is a zero of​ f(x).

12. Factor​ f(x) into linear factors given that k is a zero of​ f(x). ​f(x)=x4+3×3−42×2−172x−168​; k=−2 ​(multiplicity 2)

13. For the following polynomial​ function, find all zeros and their multiplicities.

​f(x)=

14.   Find a polynomial function​ f(x) of degree 3 with real coefficients that satisfies the following conditions.

Zero of 0 and zero of 3 having multiplicity​ 2; ​f(4​)=16

The polynomial function is ​f(x)=___________

15.   Determine the different possibilities for the numbers of​ positive, negative, and nonreal complex zeros for the following function.

​f(x)= 

What is the possible number of positive real zeros of this​ function? __________

What is the possible number of negative real zeros of this​ function? __________

What is the possible number of nonreal complex​ zeros? _____________

16.  Use​ Descartes’ rule of signs to determine the different possibilities for the numbers of​ positive, negative, and nonreal complex zeros for the following function.

​f(x)=

The possible number of positive real zeros is ______  (Type a whole number. Use a comma to separate answers as​ needed.)

The possible number of negative real zeros is ______  (Type a whole number. Use a comma to separate answers as​ needed.)

The possible number of nonreal complex zeros is _____  (Type a whole number. Use a comma to separate answers as​ needed.)

17.  Find all complex zeros of the polynomial function. Give exact values. List multiple zeros as necessary.

f(x)= 

25x−750

All complex zeros are ________

18.   Find all complex zeros of the polynomial function. Give exact values. List multiple zeros as necessary.

f(x)= 

All complex zeros are _____________