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Core Connections Integrated III, 2015
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Core Connections Integrated III, 2015 View details
1. Section 9.1
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Exercise 33 Page 443

Practice makes perfect
a
To factor the given expression, we will apply the sum of cubes. a^3+b^3=(a+b)(a^2-ab+b^2) Let's start by factoring out the GCF.
x^5+8x^2y^3
FactorOut

Factor out x^2

x^2(x^3+8y^3)
WritePow

Write as a power

x^2(x^3+2^3y^3)
ProdPow

a^m* b^m=(a * b)^m

x^2(x^3+(2y)^3)
Next, let's apply the sum of cubes formula to factor the expression further.
x^2(x^3+(2y)^3)

a^3+b^3 = (a+b)(a^2-ab+b^2)

x^2(x+2y)((x^2-x(2y)+(2y)^2)
PowProd

(a * b)^m=a^m* b^m

x^2(x+2y)((x^2-x(2y)+2^2y^2)
CalcPowProd

Calculate power and product

x^2(x+2y)(x^2-2xy+4y^2)
b
To factor the given expression, we will apply the difference of two cubes. a^3-b^3 ⇔ (a-b)(a^2+ab+b^2) Let's rewrite our expression so that it matches the above form.
8y^6-125x^3
WritePow

Write as a power

2^3y^6-3^5x^3
SplitIntoFactors

Split into factors

2^3y^(2(3))-3^5x^3
ProdInExponent

a^(m* n)=(a^m)^n

2^3(y^2)^3-3^5x^3
ProdPowII

a^m b^m = (a b)^m

(2y^2)^3-(3x)^3
Not that we have written the expression, let's use the difference of cubes formula.
(2y^2)^3-(3x)^3

a^3-b^3 = (a-b)(a^2+ab+b^2)

(2y^2-3x)( (2y^2)^2+(2y^2)(3x)+(3x)^2)
PowProdII

(a b)^m=a^m b^m

(2y^2-3x)(2^2(y^2)^2+(2y^2)(3x)+3^2x^2)
PowPow

(a^m)^n=a^(m* n)

(2y^2-3x)(2^2y^4+(2y^2)(3x)+3^2x^2)
CalcPowProd

Calculate power and product

(2y^2-3x)(4y^4+6xy^2+9x^2)
c
To factor the given expression, we will apply the difference of squares formula. a^2-b^2 ⇔ (a-b)(a+b)Let's rewrite our expression so that it matches the above form. x^6-y^6 ⇔ (x^3)^2-(y^3)^2 Not that we have written the expression, let's use the difference of squares formula. (x^3)^2-(y^3)^2 ⇕ (x^3-y^3)(x^3+y^3) Now, notice that the two factor are the difference and the sum of two cubes. Therefore, we can use the formulas we have used in Part A and Part B to simplify this expression further.
(x^3-y^3)(x^3+y^3)

a^3-b^3 = (a-b)(a^2+ab+b^2)

(x-y)(x^2+xy+y^2)(x^3+y^3)

a^3+b^3 = (a+b)(a^2-ab+b^2)

(x-y)(x^2+xy+y^2)(x+y)(x^2-xy+y^2)
Subchapter links expand_more
arrow_right 9.1.1 Introduction to Periodic Models p.433-434
3
Introduction to Periodic Models 4 (Page 433)
Introduction to Periodic Models 5 (Page 433)
Introduction to Periodic Models 6 (Page 433)
Introduction to Periodic Models 7 (Page 433)
Introduction to Periodic Models 8 (Page 434)
9
arrow_right 9.1.2 Graphing the Sine Function p.438-440
Graphing the Sine Function 13 (Page 438)
Graphing the Sine Function 14 (Page 438)
Graphing the Sine Function 15 (Page 438)
Graphing the Sine Function 16 (Page 438)
Graphing the Sine Function 17 (Page 438)
18
Graphing the Sine Function 19 (Page 439)
Graphing the Sine Function 20 (Page 439)
Graphing the Sine Function 21 (Page 439)
Graphing the Sine Function 22 (Page 439)
Graphing the Sine Function 23 (Page 440)
24
Graphing the Sine Function 25 (Page 440)
Graphing the Sine Function 26 (Page 440)
arrow_right 9.1.3 Unit Circle ↔ Graph p.442-444
Unit Circle ↔ Graph 30 (Page 442)
Unit Circle ↔ Graph 31 (Page 443)
Unit Circle ↔ Graph 32 (Page 443)
Unit Circle ↔ Graph 33 (Page 443)
Unit Circle ↔ Graph 34 (Page 443)
Unit Circle ↔ Graph 35 (Page 443)
Unit Circle ↔ Graph 36 (Page 444)
arrow_right 9.1.4 Graphing and Interpreting the Cosine Function p.447-449
Graphing and Interpreting the Cosine Function 45 (Page 447)
Graphing and Interpreting the Cosine Function 46 (Page 447)
Graphing and Interpreting the Cosine Function 47 (Page 447)
Graphing and Interpreting the Cosine Function 48 (Page 447)
Graphing and Interpreting the Cosine Function 49 (Page 447)
Graphing and Interpreting the Cosine Function 50 (Page 448)
Graphing and Interpreting the Cosine Function 51 (Page 448)
Graphing and Interpreting the Cosine Function 52 (Page 448)
Graphing and Interpreting the Cosine Function 53 (Page 448)
Graphing and Interpreting the Cosine Function 54 (Page 448)
Graphing and Interpreting the Cosine Function 55 (Page 449)
Graphing and Interpreting the Cosine Function 56 (Page 449)
Graphing and Interpreting the Cosine Function 57 (Page 449)
Graphing and Interpreting the Cosine Function 58 (Page 449)
arrow_right 9.1.5 Defining a Radian p.452-453
Defining a Radian 65 (Page 452)
Defining a Radian 66 (Page 452)
Defining a Radian 67 (Page 452)
Defining a Radian 68 (Page 452)
Defining a Radian 69 (Page 452)
Defining a Radian 70 (Page 453)
71
arrow_right 9.1.6 Building a Unit Circle p.456-457
Building a Unit Circle 76 (Page 456)
Building a Unit Circle 77 (Page 456)
Building a Unit Circle 78 (Page 456)
Building a Unit Circle 79 (Page 456)
Building a Unit Circle 80 (Page 456)
Building a Unit Circle 81 (Page 457)
Building a Unit Circle 82 (Page 457)
arrow_right 9.1.7 The Tangent Function p.460-461
The Tangent Function 88 (Page 460)
The Tangent Function 89 (Page 461)
The Tangent Function 90 (Page 461)
The Tangent Function 91 (Page 461)
The Tangent Function 92 (Page 461)
The Tangent Function 93 (Page 461)
The Tangent Function 94 (Page 461)