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Task 5

Senator Jessica Carter,

I am writing in response to the debate concerning Honors courses and their elimination. To remove the option for Honors courses would be both a mistake on the personal level, and on a national, economic level. Make no mistake, honors courses are not about boasting a superior mind or Iq. They are simply about allowing one with greater drive or ambition to exceed beyond the standard. This kind of option allows the ambitious student to go farther than he or she might have been able to without.

Concerning economics and the national level, these courses are vital. Why? Because they allow the determined, aspiring mind to be able to go farther than he or she would otherwise, thus resulting in elimination of unemployment. Not only that, but also yielding exeptional results: faster, more significant business growth, new innovation, advanced technology, better medicine, an unprecedented military, and superior international relations and influence. Enable all this to occur on a huge level, and your result is a massive, growing economy that is second to none. As you are probably thinking, Honors courses are just one small part. You are right and that they are, but the part they play is absolutely paramount. I don't think any budget cut is worth sacrificing that. Do you?

Sincerely,

Ariel Gene Fonseca

Flvs Algebra 2 Segment 1 Honors Project

Task 4

Perform the following operations and prove closure.

1. x/x+3 + x+2/x+5

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

Final:

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

Rational expression

2. x+4/x^2+5x+6 * x+3/x^2-16

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

Final:

1/(x+2)(x-4)

Rational expression

3. 2/x^2-9 - 3x/x^2-5x+6

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

Final:

(3x+4)(x+1) / (x+3)(x-3)(x-2)

Rational expression

4. x+4/x^2-5x+6 / x^2-16/x+3

(x+4)(x+3) / (x^-16)(x^2-5x+6)

Final:

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

Compare and contrast division of integers to division of rational expressions. A division of integers and a division of rational expressions are in essence the same. They have the same basic process and set up and both share the property of closure. Generally, a division problem with integers will be simpler than one with rational expressions.

Task 1

4. Find the product of (x-2i)^2.

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

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

x^2-4xi-4

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

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

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

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

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

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

x^2-(-64)

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

Check it:

(x + 8i)(x − 8i) = x^2 − 8i + 8i − 64i^2 = x^2 − 64(−1) = x^2 + 64

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

x^2-(-49)

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

Check it:

(x + 7i)(x − 7i) = x^2 − 7i + 7i − 49i^2 = x^2 − 49(−1) = x^2 + 49

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

(x + 9i)^2

x^2 + 2(x • 9i) + (9i)^2

x^2 + 18xi + 9^2i^2

x^2 + 18xi + 81(−1)

x^2 + 18xi − 81

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:

4 roots, 2 real, 2 complex: 2, -2, 5i, -5i

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

3 roots: 2 real: 5 (multiplicity of 2), -5

The following graph shows a seventh-degree polynomial:

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

-5, -1, 4, 7

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

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 2

(2x - 3y)^4

(2x-3y)(2x-3y)(2x-3y)(2x-3y)

Expand

Final Solution

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

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

The following are possibilities becuause they have the correct variables and their degrees are 8.

a^5b^3, a^4b^4, ab^7

Expand the following using the Binomial Theorem and Pascal’s triangle.

(x + 2)^6

(x+2)(x+2)(x+2)(x+2)(x+2)(x+2)

Expand

Final solution

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

(x - 4)^4

(x-4)(x-4)(x-4)(x-4)

Expand

Final Solution

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

(2x + 3)^5

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

Expand

Final Solution

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

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