Polynomial identities to complex numbers:

Task 2

Expanding the following using binomial theorem and Pascal's triangles:

(x+2)^6: I get the answer x^6+12x^5+60x^4+160x^3+240x^2+192x+64 when I expand the binomial theorem.

(x-4)^4: I get the x^4-16x^3+96x^2-256x+256 when I expand the binomial theorem.

(2x+3)^5: I get the 32x^5+240x^4+720x^3+1080x^2+810x+243.

(2x-3y)^4: I get the 16x^4-96x^3y+216x^2y^216xy^3+81y^4 when I expand the binomial theorem.

In the expression of (3a+4b)8, which of the following are positive variables terms? Explain your reasoning. a^2b^3; a^5b^3; ab^8; a^4b^4; a^8; ab^7; a^6b^5: I picked a^5b^3, ab^7, a^4b^4, a^8, and b^8 because they all add up to 8.

Task 3

Using the fundamentals of algebra, complete the following:

Determine how many, what type, and the root for f(x)=x^4+21x^2-100: I found 2 complex and 2 real roots. They are x=2, x=-2, x=5i, and x=-5i.

Determine how many, what type, and the root for f(x)=x^3-5x^2-25x+125: I found 2 real roots. They are x=5 and x=-5.

The following graph shows a seventh-degree polynomial:

Part 1: List the polynomial's zeros with possible multiplicities: The polynomial's zeros are (-5,0), (-1,0), (4,0), (7,0).

Part 2: Write a possible factored form of the seventh degree function: f(x)=(x+5)^4 (x+1) (x-4) (x-7).

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

Perform the following operations and prove closure:

1. (x/x+3)+(x+2/x+5)= The closure of the operations is 2(x^2+5x+3)/(x+3)(x+5).

2. (x+4/x^2+5x+6) * (x+3/x^2-16)= The closure of the operation is 1/(x+2)(x-4).

3. (2/x^2-9)-(3x/x^2-5x+6)= The closure of the operation is 1(3x+4)(x+1)/(x-2)(x+3)(x-3).

4. (x+4/x^2-5x+6)/(x^2-16/x+3)= The closure of the operation is x+3/(x-2)(x-3)(x-4).

Compare and contrast division of the integers to division of rational expression:

The division of integers is division of whole numbers without any variables. The division of rational expression is division of integers or making fractions with no variables.

Task 5

GOAL!

**05.10 Segment one Honors project**

**Vineeth Munigati**

Factor x^2+64: First we convert (x^2+64) to (x^2-64) since (x^2+64) is supposed to convert polynomials function to complex numbers. So, we get (x-8)(x+8).

Factor 16x^2+64: First we convert (16x^2+64) to (16x^2-64) since (16x^2+64) is supposed to convert polynomial function to complex numbers. So, we get (4x^2-8)(4x^2+8).

Find the product of (x+9i)^2: I got the answer x^2+18ix-81 by un-factoring the (x+91)^2.

Find the product of (x-2i)^2: I got the answer x^2-4ix-4 by un-factoring the (x-2i)^2.

Find the product of (x+(3+5i))^2: I got the answer x^2+6x+9 by expanding the binomial when multiplying.

Write a letter or create a presentation to Senator Jessica Carter:

-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 spend to subsidize it?

Dear, Senator Jessica Carter.

I think the honors standards are necessary because it gives more challenging work for the advanced kids who are highly motivated and capable of doing intensive work than regular standards which enhances their mental ability. So, yes I feel that honors standards are really necessary. As honors standards are really challenging and intensive, it enables us to understand things deeply in real world. Irrespective of the fiscal position of any country, I highly support the endorsement and subsidising honors programs in public schools because most of the families cannot afford to pay for private education for high standards which helps the advanced kids think more critically and highly motivate them and use their brains for the development of the nation in all the fields.

Sincerely,

Vineeth Munigait.