# Applying Polynomial Identities to Complex Numbers Worksheet

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User Generated
Subject
Mathematics
Type
Worksheet
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Segment 1 Honors Project
By Elmarie Y. Reyes
Complete the following exercises by applying polynomial identities to complex numbers.
1. Factor x
2
(x-8i)(x+8i)
x2 8ix + 8ix - 64i2
x2 +64
2. Factor 16x
2
(4x-7i)(4x+7i)
16x2 28ix + 28ix 49i2
16x2 + 49
3. Find the product of (x + 9i)
2
.
(x+9i)(x+9i)
x2 + 9ix + 9ix +81i2
x2 + 18ix - 81
4. Find the product of (x − 2i)
2
.
(x-2i)(x-2i)
x2 2ix 2ix +4i2
x2 -4xi -4
5. Find the product of (x + (3+5i))
2
.
(x+(3+5i)(x+(3+5i)
x2 + 3x + 5ix + 3x + 5ix + 9 + 15i + 15i + 25i2
x2 + 6x + 30i + 10ix + 9 -25
x2 + 6x + 30i + 10ix -16

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Expand the following using the Binomial Theorem and Pascal’s triangle.
1. (x + 2)
6
(x+2)(x+2)=x
2
+4x+4
(x
2
+4x+4)(x+2)=x
3
+6x
2
+12x+8
(x
3
+6x
2
+12x+8)(x+2)=x
4
+8x
3
+24x
2
+32x+16
(x4+8x3+24x2+32x+16)(x+2)=x
5
+10x
4
+40x
3
+80x
2
+80x+32
(x5+10x4+40x3+80x2+80x+32)(x+2) =x
6
+12x
5
+60x
4
+160x
3
+240x
2
+192x+64
x6+12x5+60x4+160x3+240x2+192x+64
2. (x − 4)
4
(x-4)(x-4)=x
2
-8x+16
(x
2
-8x+16)(x-4)=x
3
-12x
2
+48x-64
(x
3
-12x
2
+48x-64)(x-4)=x
4
-16x
3
+96x
2
-256x+256
x
4
-16x
3
+96x
2
-256x+256
3. (2x + 3)
5
(2x+3)(2x+3)=4x
2
+12x+9
(4x
2
+12x+9)(2x+3)=8x
3
+36x
2
+54x+27
(8x
3
+36x
2
+54x+27)(2x+3)=16x
4
+96x
3
+216x
2
+216x+81
(16x
4
+96x
3
+216x
2
+216x+81)(2x+3)=32x
5
+240x
4
+720x
3
+1080x
2
+810x+243
32x
5
+240x
4
+720x
3
+1080x
2
+810x+243
4. (2x − 3y)
4
(2x-3y)(2x-3y)=4x
2
-12xy+9y
2
(4x
2
-12xy+9y
2
)(2x-3y)=8x
3
-36x
2
y+54xy
2
-27y
3
(8x
3
-36x
2
y+54xy
2
-27y
3
)(2x-3y)=16x
4
-96x
3
y+216x
2
y
2
-216xy
3
+81y
4
16x
4
-96x
3
y+216x
2
y
2
-216xy
3
+81y
4
5. In the expansion of (3a + 4b)
8
, which of the following are possible variable terms? Explain your
reasoning. a
2
b
3
; a
5
b
3
; ab
8
; b
8
; a
4
b
4
; a
8
; ab
7
; a
6
b
5
The terms that can be possible answers for (3a + 4b)
8
are: a
8
,a
5
b
3
,a
4
b
4
,ab
7
, and b
8
. I got this by
multiplying my variables, I knew that if I kept multiplying my numbers it would give me a lot of
numbers, so I decided to get my numbers out and just multiply (a + b) 8 times since I was only
asked to find the variable terms.
E.J.
(a+b)(a+b)
(a
2
+2ab+b
2
)(a+b)
a
3+
3a
3
b+3ab
2
+b
3
and so on until I got my final answer

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Segment 1 Honors Project By Elmarie Y. Reyes Task #1 Complete the following exercises by applying polynomial identities to complex numbers. 1. Factor x2 + 64. Check your work. (x-8i)(x+8i) x2 – 8ix + 8ix - 64i2 x2 +64 2. Factor 16x2 + 49. Check your work. (4x-7i)(4x+7i) 16x2 – 28ix + 28ix – 49i2 16x2 + 49 3. Find the product of (x + 9i)2. (x+9i)(x+9i) x2 + 9ix + 9ix +81i2 x2 + 18ix - 81 4. Find the product of (x − 2i)2. (x-2i)(x-2i) x2 – 2ix – 2ix +4i2 x2 -4xi -4 5. Find the product of (x + (3+5i))2. (x+(3+5i)(x+(3+5i) x2 + 3x + 5ix + 3x + 5ix + 9 + 15i + 15i + 25i2 x2 + 6x + 30i + 10ix + 9 -25 x2 + 6x + 30i + 10ix -16 Task #2 Expand the following using the Binomial Theorem and Pascal’s triangle. 1. (x + 2)6 (x+2)(x+2)=x2+4x+4 (x2+4x+4)(x+2)=x3+6x2+12x+8 (x3+6x2+12x+8)(x+2)=x4+8x3+24x2+32x+16 (x4+8x3+24x2+32x+16)(x+2)=x5+10x4+40x3+80x2+80x+32 (x5+10x4+40x3+80x2+80x+32)(x+2) =x6+12x5+60x4+160x3+240x2+192x+64 Final answer is: x6+12x5+60x4+160x3+240x2+192x+64 2. (x − 4)4 (x-4)(x-4)=x2-8x+16 (x2-8x+16)(x-4)=x3-12x2+48x-64 (x3-12x2+48x-64)(x-4)=x4-16x3+96x2-256x+256 Final answer is: x4-16x3+96x2-256x+256 3. (2x + 3)5 (2x+3)(2x+3)=4x2+12x+9 (4x2+12x+9)(2x+3)=8x3+36x2+54x+27 (8x3+36x2+54x+27)(2x+3)=16x4+96x3+216x2+216x+81 (16x4+96x3+216x2+216x+81)(2x+3)=32x5+240x4+720x3+1080x2+810x+243 Final answer is: 32x5+240x4+720x3+1080x2+810x+243 4. (2x − 3y)4 (2x-3y)(2x-3y)=4x2-12xy+9y2 (4x2-12xy+9y2)(2x-3y)=8x3-36x2y+54xy2-27y3 (8x3-36x2y+54xy2-27y3)(2x-3y)=16x4-96x3y+21 ...
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Excellent resource! Really helped me get the gist of things.

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