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 One

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

Task Two

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

Task Three

Using the Fundamental Theorem of Algebra, complete the following:

Task Four

Perform the following operations and prove closure.

Task Five

Senator Jessica Carter,

Regarding the matter of honors courses in schools, I feel that they are vital to every student's education. They are necessary for students to hold themselves to a standard, and give opportunities to bright young adults to be challenged. The educational system is already corrupt in that curriculum and teaching styles are not reaching every student to the level which is necessary. The use of these honors ideals and applications is allowing students to succeed in ways they would not otherwise be able to. The concepts I exemplified in this presentation show how important they are in the real world. I believe that there are many uses for these applications in engineering, accounting, and many other careers. Honors programs prepare adolescents for the real world, and that should be a priority when allocating funding in the educational system.

**5.10 Honors Project**

**Hannah Terzi**

Segment One

Segment One

Essential Questions

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

To factor this, a polynomial identity must be applied, i.e the Difference of Squares. How? Change the equation to subtraction. x^2 - (-64) and then expand it to (x + 8i)(x- 8i)

Checking it is as easy as applying it to the Sum of Squares :

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

and plugging it back in to prove that (x + 8i)(x – 8i) = x^2 – 8i + 8i – 64i^2 = x^2 – 64(-1) = x^2 + 64

Where * = Square root symbol

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

This expression is similar to the one above in that the same identities can be used.

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

(4x + 7i)(4x – 7i) = 16x^2 – 7i + 7i – 49i^2 = 16x^2 – 49(-1) = 16x^2 + 49

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

(x + 9i)^2

x^2 + 2(x • 9i) + (5i)^2

x^2 + 18xi + 81i^2

x^2 + 18xi – 81

4. Find the product of (x – 2i)^2

(x – 2i)^2

x^2 + 2(x • –2i) + (–2i)^2

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

x^2 – 4xi – 4

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

(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

1. (x + 2)^6

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

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.

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

The only terms that are possible in that particular expansion would be:

a^5b^3; ab^8; b^8; a^4b^4; a^8; ab^7 because the exponents add to 8 on these terms.

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

Factored, it is (x+2)(x-2)(x^2+25) or (x+2)(x-2)(x+5i)(x-5i)

How many: 4, since the polynomial is to the power of 4.

What type: Two real roots and two complex roots.

The roots are: x+2 = 0, x= -2

x-2=0, x= 2

x-5i=0, x=5i

x+5i=0, x=-5i

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

Factored, it is (x+5)(x-5)^2

How many: 3, since the polynomial is to the power of 3.

What type: Two real roots

The roots are: x - 5=0, x=5 with a multiplicity of 2.

x+5 = 0, x= -5

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

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

x = -5, multiplicity of 2

x = -1, multiplicity of 1

x = -2, multiplicity of 1

x = 4, multiplicity of 1

x = 7, multiplicity of 1

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

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

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.

2.

3.

2(x^2+5x+3)/(x+3)(x+5), x cannot equal -3 or -5

1/(x+2)(x-4), x cannot equal -2,-3,4, or -4

- 1(3x+4)(x+1)/(x+3)(x-3)(x-2), x cannot equal 2,3 or -3

4. {(x+4)/x^2-5x+6}/{x^2-16/x+3} = x+3/(x-2)(x-3)(x-4)

x cannot equal 2, 3 or -3

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

Though the two processes are quite similar in that they both require closure, I find that dividing integers is much easier. Rational expressions require much more focus and meticulousness because they are longer and harder.

Thank you for watching!