1.

f(x)=x^4+21x^2-100

(x-2)(x+2)(x+5i)(x-5i)

x=2 ,x=-2 ,x=5 ,x=-5

4 roots. 2 real, 2 complex.

2.

f(x)=x^3-5x^2-25x+125

(x-5)(x^2-25x+125)

x=5 ,x=-25+-/-2

3 roots. 1 real, 2 complex.

3. Part1:

(-1,0)(-5,0)^4(4,0)(7,0)

Part2:

F(x):(x+5)^4(x+1)(x-4)(x-7)

4.

f(x)=(x+8)^3(x+6)^2(x+2)(x-1)^3(x-3)^4(x-6)

Task 4 Questions:

Perform the following operations and prove closure.

1.x over x plus 3 plus x plus 2 over x plus 5

2.x plus 4 over x squared plus 5x plus 6 times

x plus 3 over x squared minus 16.

3.2 over x squared minus 9 minus 3x over x squared 5x plus 6

4.

x+4

x2−5x+6

÷

x2−16

x+3

5.Compare and contrast division of integers to division

of rational expressions.

Task 4 Answers:

Task 1 Answers:

1.

x^2 +64

(x+8i)(x-8i)

Check work:

x^2-8xi+8xi-64i^2

x^2+64

2.

16x^2+49

(4x+7i)(4x-7i)

check work:

16x^2-28xi+28xi-49i^2

16x^2+49

Task 2 Answers:

1.

(x+2)^6

=x^6+6x^5 2+15x^4 2^2+20x^3 2^3+15x^2 2^4+6x 2^5+2^6

=x^6+6*2x^5+15*4x^4+20*8x^3+15*16x6^2+6*32x+64

=x^6+12x^5+60x^4+160x^3+240^2+192x+64

2.

(x-4)^4

=x^4+4x^3-4+6x^2-4^2+4x-4^3+-4^4

=x^4+4*-4x^3+6*16x^2+4*-64x-256

=x^4-16x^3+96x^2-256x-256

3.

(2x+3)^5

=2x^5+10x^4 3^1+20x^3 3^2+20x^2 x63+10x3^4+3^5

=2x^5+10*3x^4+20*9x^3+20*27x^2+10*81x+243

=2x^5+30x^4+180x^3+540x^2+810x+243

4.

(2x-3y)^4

=2x^4+8x^3-3y+12x^2-3y^2+8x-3y^3+-3y^4

=2x^4-24yx^3+108yx^2-216yx+81

5.

(3a+4b)^8

Because the A's degree is always decreasing, the B's

degree is always increasing.

Task 1

Questions:

Complete the following exercises by applying polynomial identities to complex numbers.

1.Factor x2 + 64. Check your work.

2.Factor 16x2 + 49. Check your work.

3.Find the product of (x + 9i)2.

4.Find the product of (x – 2i)2.

5.Find the product of (x + (3+5i))2.

**05.10 Segment One Honors Project**

Task 2 Questions:

Expand the following using the binomial theorem and Pascal’s triangle.

1.(x + 2)6

2.(x – 4)4

3.(2x + 3)5

4.(2x – 3y)4

5.In the expansion of (3a + 4b)8, which of the following are possible variable terms? Explain your reasoning.

a^2b^3; a^5b^3; ab^8; b^8; a^4b^4; a^8; ab^7; a^6b^5

Task 3 Questions:

Using the fundamental theorem of

algebra, complete the following:

1.Determine how many, what type, and find the roots

for f(x) = x4 + 21x2 – 100.

2.Determine how many, what type, and find the roots

for f(x) = x3 - 5x2 – 25x + 125.

3.The following graph shows a seventh-degree polynomial:

graph of a polynomial that touches the x axis at negative 5,

crosses the x axis at negative 1, crosses the y axis at negative 2,

crosses the x axis at 4, and crosses the x axis at 7.

Part 1: List the polynomial’s zeroes with possible multiplicities.

Part 2: Write a possible factored form of the seventh degree function.

4.Without plotting any points other than intercepts, draw a

possible graph of the following polynomial:

f(x) = (x + 8)3(x + 6)2(x + 2)(x – 1)3(x – 3)4(x – 6).

1

. x + x+2

x+3 x+5

(x+5)(x)+(x+3)(x+2)

(x+3)(x+5)

x^2+5x+x^2+5x+6

x^2+8x+15

2x^+10x+6

x^2+8x+15

2(x^2+5x+3)

(x+3)(x+5)

2.

x+4 * x+3

x^2+5x+6 x^2-16

(x+4)(x+3)

(x+3)(x+2)(x+4)(x-4)

1

(x+2)(x-4)

3.

(x+9i)^2

(x+9i)(x+9i)

x^2+9xi+9xi+81i^2

x^2+18xi-81

4.

(x-2i)^2

(x-2i)(x-2i)

x^2-2xi-2xi+4i^2

x^2-4xi-4

5.

(x+(3+5i))^2

(x+3+5i)(x+3+5i)

x^2+3x+5xi+3x+9+15i+5xi+15i+25i^2

x^2+6x+10xi+9+30i+25i^2

x^2=6x+10xi-16+30i

3.

2 - 3x

x^2-9 x^2-5x+6

(2)(x^2-5x+6)-(3x)(x^2-9)

(x^2-9)(x^2-5x+6)

(2x^2-10x+12)-(3x^3-27x)

(x^2-9)(x^2-5x+6)

-3x^3+2x^2-17x+12

(x^2-9)(x^2-5x+6)

1(3x+4)(x+1)

(x+3)(x-3)(x-2)

Continued...

Task 4

Answers Part 2:

4.

x+4 / x^2-16

x^2-5x+6 x+3

(x+4)(x+3)

(x^2-16)(x^2-5x+6)

(x+4)(x+3)

(x+4)(x-4)(x-2)(x-3)

x+3

(x-4)(x-2)(x-3)

5.Division of intergers is basically a fraction.

Ex: 1/2 or 8/3

Division of rational expressions is a polynomial over a polynomial.

Ex: x^2+15

x+3

Task 5 Question:

Write a letter or create a presentation for Senator Jessica Carter.

Your task is to either convince her that Algebra 2 Honors is

necessary and important to advanced students or to advise her that funding should be spent elsewhere. Be sure to address the following questions:

•Are Honors standards really necessary?

•How are the Honors standards from this lesson used in the real world?

•Is the Honors endorsement valuable enough that scarce educational funding should be spent to subsidize it?

Be sure to include evaluations as to the importance (or non importance) of each of the four standards covered in this lesson and include real-world examples and applications as appropriate to strengthen your argument.

Task 5 Answer:

Dear Seantor Jessica Carter,

An honor course is a class where students who

feel like they need a challenge are located. Majority of the students placed in honors courses feel that regular classes are too easy. While some students prefer no challenges, others love challenges. I believe honors standards are necessary because it sets specific standard for students to reach, making them push to achieve. The honor standards from this lesson are used in the real world in many ways. For example, when cooking, you use math to measure the substances needed to make the product. Also, scientists use math to measure chemicals in the lab. The honors endorsement is valuable enough that scarce educational funding should be spent on the program becuase without the funds there would be no honors classes available. Which means those students who focus on their education more than others would have no challenges and be forced to be in classes with students who don't care as much.