Chase Williams

Ms. Haramis

Task 1 Q&A

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

1. Factor x2 + 64. Check your work.

2. Factor 16x2 + 49. Check your work.

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

4. Find the product of (x − 2i)2.

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

Answers

1. x^2 +64=

Answer: (x+8i)(x-8i)

2. 16x^2+49=

Answer: (4x+7i)(4x-7i)

3. (x+9i)^2= (x+9i)(x+9i= x^2+9ix+9ix+81i^2=x^2+18ix+(-81)=

Answer: x^2+18ix-81

4. (x-2i)^2=(x-2i)(x-2i)=x^2-2ix-2ix+4i^2=x^2-4ix+(-4)=

Answer: x^2-4ix-4

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

X^2+3x+5ix+3x+9+15i+5ix+15i+25i^2=x^2+6x+10ix+30i+25i^2+9=

Answer: X^2+6x+10ix+30i-25+9= x^2+6x+10ix+30i-16

Task 2 Q&A

Expand the following using the Binomial Theorem and
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(x + 2)6

2. (x − 4)4

3. (2x + 3)5

4. (2x − 3y)4

5. 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

Answers

1. (x +
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(2x − 3y)^4=16x^4-96x^3y+216x^2y^2-216xy3+81y^4

5. The possible variable terms would have to be a2b3; a5b3; ab8; b8; a4b4; a8; ab7; a6b5 because the exponents of each one of the terms all add up to 8.

Task 3 Q&A

Using the Fundamental Theorem of Algebra, complete the following:

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

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

3. 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.

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).

Answers

1. There will be 4 roots total, Two real and two complex, and the roots are x=-2, x=2, x=5i, x=-5i

2. There will be 3 roots total, two real, and the roots are x=5 with a multiplicity of 2, and