**5.10 Honors Project-Algebra 2**

Jessica Sweigard

Jessica Sweigard

Using the Fundamental Theorem of Algebra, complete the following:

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

First factor: (x+2)(x-2)(x+5i)(x-5i)

There are 4 roots, because that's the number the polynomial is powered. 2 roots are real, and 2 are complex. The roots are: x= -2; x= 2; x=5i; and x=-5i.

Perform the following operations and prove closure.

Task 1

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

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

1. (x+2)^6

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

**Task 5**

1. (x^2+64) (x+8i)(x-8i)

(x^2-(-64) x^2+8ix-8ix-64i^2

(x+8i)(x-8i) (x^2+64)

2. 16x^2+49 (4x+7i)(4x-7i)

(16x^2-(-49) 16^2+7ix-7ix-49i^2

(4x+7i)(4x-7i) 16x^2+49

3. (x + 9i)^2 x^2+18xi–81

x^2+2(x*9i)+(9i)^2 (x+9i)(x+9i)

x^2+18xi+81i^2 (x+9)^2

x^2+18xi-81

4. (x-2i)^2 x^2 – 4xi – 4

x^2+2(x*–2i)+(-2i)^2 (x-2i)(x-2i)

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

x^2–4xi–4

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

x^2 + 2(x • (3+5i)) + (3+5i)^2

x^2 + 2(3x+5xi) + ( 9 + 30i + 25i^2)

x^2 + 6x + 10xi + 9 + 30i – 25

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

Task 2

2. (x-4)^4

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

3. (2x+3)^5

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

4. (2x – 3y)^4

16x^4-96x^3y+216x^2y^2-216xy3+81y^4

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

The only terms you can expand are a^5b^3; ab^8; b^8; a^4b^4; a^8; ab^7 because 8 is added to the exponents of each of them.

So the final answer is: a^2b^3; a^5b^3; ab^8; b^8; a^4b^4; a^8; ab^7; ^b^5

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

Factor:(x+5)(x-5)^2

There are 3 roots, because 3 of the power of the polynomial. There are 2 real roots: x=-5 and x=5.

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

Part 1: The polynomial zeroes could be: Part 2: Possible factored form:

x=-5, multiplicity of 2

x=-1, multiplicity of 1

x= 4, multiplicity of 1

x= 7, multipliciy of 1

x^2-x^5+x^3

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). We can get the possible zeroes from this: x=-8, x=-6, x=-2, x=1, x=3, and x=6.

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

x can't = -5, or -3.

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

x can't = -2, or -4.

1.

2.

3.

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

x can't =-3,2, or 3.

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

Dividing integers is much simpler than dividing rational expressions because there are no variables, and not as many steps.

4. x+4/x^2−5x+6 ÷ x^2-16/x+3

x+3/(x-2)(x-3)(x-4) x can't = 2, 3, or 4.

Dear Senator Jessica Carter,

Being the Honors student that I am, I strongly believe that Honors classes are completely necessary in unlocking a students full potential for their future. I have maintained good grades in honors classes since elementary school, and now I'm enrolling into a dual enrollment program to further my education. There was one class I took last year that wasn't an honors class, and that was my Geometry class. It was a very slow paced class, and I was extremely bored during most of the class, and I still finished with an A. The 4 standards in this lesson can help you understand that there is a way to take difficult problems and find solutions. In order to have challenging classes for Honors students means ofcourse having to buy school books, calculators, and other materials, but it won't be a waste of funding because those students are learning important skills.