**Segment 1 Honors Project**

**Emily Jones**

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 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.

Essential Questions

How do polynomial identities apply to complex numbers?

How does the binomial theorem use Pascal’s triangle to expand binomials raised to positive integer powers?

How is the fundamental theorem of algebra true for quadratic polynomials?

Are rational expressions closed under addition, subtraction, multiplication, and division?

Task 2

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

(x + 2)^6

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

(x – 4)^4

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

(2x + 3)^5

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

(2x – 3y)^4

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

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

ab^7, a^5b^3, a^4b^4, a^8, b^8 would be a possible variable term because the degrees add up to 8.

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

Task 3

Using the fundamental theorem of algebra, complete the following:

Determine how many, what type, and find the roots for f(x) = x^4 + 21x^2 – 100.

How many: 4 (since degree is 4)

Types: 2 real, 2 complex

Roots: +-2, +-5i

Determine how many, what type, and find the roots for f(x) = x^3 - 5x^2 – 25x + 125.

How many: 3

Types: 2 real

Roots: +-5

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.

The multiplicity of -5 should have an even multiplicity because it touches the graph but doesn't cross it. -1, -2, -4, and 7 odd multiplicity because they cross the x axis.

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

(x+5)^2(x-7)(x+4)(x+2)(x+1)

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 1

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

Factor x^2 + 64. Check your work.

(x+8)(x+8)

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

(4x+7)(4x+7)

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

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

x^2+18i+81i^2

x^2+18i-81

Find the product of (x – 2i)^2.

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

x^2-4i+4i^2

x^2-4i-4

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

(x+3+5i)^2

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

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

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

Task 4

Perform the following operations and prove closure.

1.

x(x+5)+(x+2)(x+3) x^2+5x+x^2+5x+6 2x^2+10x+6

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

2.

x+4 x+3 1

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

3.

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

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

4. (x+4)/

(x^2−5x+6) x+4 x+3 x+3

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

(x^2−16)/

(x+3)

Compare and contrast division of integers to division of rational expressions.

The division of integers and division of rational expressions but require you to use KCF, but with rational expressions they involve a variable while division of integers does not.

Dear Ms. Jessica Carter,

It is my duty to represent all honors students and inform you of the importance of keeping Algebra 2 Honors is necessary. Honors is important because they allow already advanced students to challenge themselves and keep their minds sharp instead of sitting in a classroom bored because they've already completed all the work. Also, these four standards in this lesson can be applied to real life because they can and will be used in future jobs that use math like quite simply a math teacher or an accountant, and, without even realizing it, we use these standards and methods taught in Algebra 2 Honors all the time. Physicists and chemists or engineers will use these especially. So the fact of it is that Algebra 2 Honors is necessary and funding needs to spent for it.