**Task 1**

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

1. Factor x^2 + 64. Check your work.

I applied the polynomial identity by subtracting the operation instead of adding. When converted to x^2-64 it = (x-8)(x+8)

2. Factor 16x^2 + 49. Check your work.

I applied the polynomial identity by subtracting much like the last problem, then found the difference of squares. Once I changed the polynomial identity it was (4x-7)(4x+7)

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

I found the product by using the sum of squares, this gave me the answer of x^2+18ix-81

4. Find the product of (x – 2i)^2.

I found the product by expanding the binomial, this then gave me the answer x^2-4ix-4

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

I found the product by expanding the binomial much like the last problem. After multiplying this gave me the answer of x^2+6x+9

Task 2

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

1. (x + 2)6

When expanded using the binomial theorem I ended up with the solution of:

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

2. (x – 4)4

When expanded using the binomial theorem I ended up with the solution of:

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

3. (2x + 3)5

When expanded using the binomial theorem I ended up with the solution of:

32x^5+240x^4+720x^3+1080x^2+810x+243

4. (2x – 3y)4

When expanded using the binomial theorem I ended up with the solution of:

16x^4-96x^3y+216x^2y^2-216xy^3+81y^4

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

a2b3; a5b3; ab8; b8; a4b4; a8; ab7; a6b5

The degree of the term must add up to 8. ab^7 a degree of 8. This is because ab^7 is the same as a^1 * b^7. Add the 1 and the 7 exponents and you get 8.

Task 3

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.

The roots for this operation are (x^2+25)(x^2-4)

2. Determine how many, what type, and find the roots for f(x) = x3 - 5x2 – 25x + 125.

The roots for this operation are (x-5)^2(x+5)

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

*couldn't transfer the graph

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

This particular polynomial has both even and odd multiplicities. This is because the function crosses the x-axis at the root, and the even function touches the graph at the root. The polynomial zeroes are -5, -2, 3, 7

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

The degree of the polynomial is 7. The possible factored form is (x^2+25)(x-2)(x+2)(x-5)^2(x+5).

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).

Task 4

Task 5

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 rally necessary? Yes, I believe that honors standards are necessary. It sets a standard that students can strive for, giving extra incentive to do better. It also alows children that are already at that level to extend their knowledge even more.

How are the Honors standards from this lesson used in the real world? The honors standards from not only this lesson but in all the rest are used on many occasions in the real world. Algebra is used everyday whether it be simple tasks or complicated. Algebra teaches you how to problem solve, and can help with something as common as figuring out how to pay bills.

Is the Honors endorsement valuable enough that scarce educational funding should be spent to subsidize it? Yes, as I stated earlier Algebra is something fundamental to everyday life and can help with both simple and complicated problems we may have. Honors simply builds upon that and allows for even greater growth for todays students.

Perform the following operations and prove closure.

1. For this operation the closure is 2(x^2+5x+3)(x+3)(x+5)

2. For this operation the closure is 1/(x+2)(x-4)

3. For this operation the closure is

-3x^3+2x^2-17x+12/(x^2-9)(x^2-5x+6)

4. For this operation the closure is

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

5. The divisions of integers compared to rational expressions for me is the easiest to use. In order to divide you would have to first determine the quotient.