Using the Fundamental Theorem of Algebra, complete the following:

Determine how many, what type, and find the roots for f(x) = x4 + 21x2 − 100.

the roots are - x+2=0 , x=-2

x-2=0 , x=2

x-5i=0, x=5i

x+5i=0 ,x=-5

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.

Determine how many, what type, and find the roots for f(x) = x3 − 5x2 − 25x + 125.

The following graph shows a seventh-degree polynomial:

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

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

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

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)

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). {(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

Task 1

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

Factor x2 + 64. Check your work.

Factor 16x2 + 49. Check your work.

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

Find the product of (x − 2i)2.

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

Task 1 Answers

Factor x2 + 64.

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

Factor 16x2 + 49.

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

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

x*2+18xi-81

Find the product of (x − 2i)2.

x*2-4xi-4

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

x*2+6+10xi+30i

Task 3

Using the Fundamental Theorem of Algebra, complete the following:

Determine how many, what type, and find the roots for f(x) = x4 + 21x2 − 100.

Determine how many, what type, and find the roots for f(x) = x3 − 5x2 − 25x + 125.

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.

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

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

(x + 2)6

(x − 4)4

(2x + 3)5

(2x − 3y)4

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

a2b3; a5b3; ab8; b8; a4b4; a8; ab7; a6b5

Task 2 Answers

(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

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

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

a2b3; a5b3; ab8; b8; a4b4; a8; ab7; a6b

the only terms that are possible in this problem would be - a*5 b*3; ab*8; b*8 ;a* 4 b*4

**05.10 Segment One Honors Project**

Alyssa Stephens

Alyssa Stephens

Task 4

Perform the following operations and prove closure.

x over x plus 3 plus x plus 2 over x plus 5

x plus 4 over x squared plus 5x plus 6 times x plus 3 over x squared minus 16.

2 over x squared minus 9 minus 3x over x squared 5x plus 6

x+4x2−5x+6 ÷ x2−16x+3

Compare and contrast division of integers to division of rational expressions

Task 4 answers

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.

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Task 5 answer

x over x plus 3 plus x plus 2 over x plus 5

x plus 4 over x squared plus 5x plus 6 times x plus 3 over x squared minus 16.

1. 2(x^2+5x+3)/(x+3)(x+5), x cannot equal -3 or -5

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

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

2 over x squared minus 9 minus 3x over x squared 5x plus 6x+4x2−5x+6 ÷ x2−16x+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

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

Though the two processes are alike and they both need closure. Rational expressions are simply fractions containing a polynomial in the numerator and denominator. They will still work the same way fractions work. So if you find yourself stuck, think about how you would simplify or solve using fractions and apply the same steps.

Integers - To multiply or divide signed integers, always multiply or divide the absolute values and use these rules to determine the sign of the answer:

The product of two positive integers or two negative integers is positive.

The product of a positive integer and a negative integer is negative.

Dear Senator Jessica Carter,

There are three questions about Algebra 1 and 2 honors.

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?

These problem-solving and critical-thinking

skills can help students succeed in their jobs

and their lives even if they do not continue

their education beyond high school or do not

pursue a math- or science career.

Algebra is important for future employment

opportunities . Researchers have found that the higher the level of math courses students take

in high school, the greater chance those

students will attend and graduate from

college and find better paying jobs.

Students who pass Algebra 1 in 8th or 9th

grade have more academic options. Besides

meeting the basic requirement for high

school graduation. they have completed Algebra I before taking the florida High School Exit Exam in

10th grade, making them better prepared

to pass the math section of that exam. All these are reasons are the best reasons to take Algerbra 1 and 2 honors.

THANKS FOR WATCHING!!