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Home » Algebra » Simple Formulae and their Application

Simple Formulae and their Application

In the article Simple Formula and their Applications I we dealt with algebraic formulas in the second degree, i.e., formulas related to perfect squares and the sum and difference of two squares. In this article we will be covering the algebraic formulas in the third degree, i.e., formulas related to perfect cubes and the sum and difference of two cubes.

Formula:

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

Again, (a+b)^3=a^3+b^3+3ab(a+b) [Taking 3ab common from 3a^2b+3ab^2 ]

Proof:

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

Corollary: a^3+b^3=[a^3+b^3+3ab(a+b)]-3ab(a+b)

or, a^3+b^3=(a+b)^3-3ab(a+b)

Example: Find the cube of 3a+5b

Solution:

(3a+5b)^3=(3a)^3+3(3a)^2(5b)+3(3a)(5b)^2+(5b)^3 \\    =27a^3+3(9a^2)(5b)+3(3a)(25b^2)+125b^3 \\    =27a^3+135a^2b+225ab^2+125b^3

Example: Find the cube of p+q+r

Solution:

(p+q+r)^3=[(p+q)^+r]^3=(p+q)^3+3(p+q)^2r+3(p+q)r^2+r^3 \\    =(p^3+3p^2q+3pq^2+q^3)+3(p^2+q^2+2pq)r+3(p+q)r^2+r^3 \\    =p^3+q^3+r^3+3pq^2+3p^2r+3pr^2+3q^2r+3qr^2+6pqr

Example: If x+\dfrac{1}{x}=p , show that x^3+\dfrac{1}{x^3}=p^3-3p

Solution:

\because a^3+b^3=(a+b)^3-3ab(a+b)

\therefore x^3+\dfrac{1}{x^3}=( x+\dfrac{1}{x})^3-3.x.\dfrac{1}{x}.( x+\dfrac{1}{x}) \\    =(x+\dfrac{1}{x})^3-3(x+\dfrac{1}{x})=p^3-3p [\because x+\dfrac{1}{x}=p ] (Proved)

Example: Simplify (x-y)^3+(x+y)^3+3(x-y)^2(x+y)+3(x+y)^2(x-y)

Solution:

Putting a for x-y and b for x+y we have,

The given expression =a^3+b^3+3a^b+3ab^2

= a^3+3a^2b+3ab^2+b^3 \\    =(a+b)^3=[(x-y)+(x+y)]^3=(2x)^3=8x^3

Example: Show that x^3+\dfrac{1}{x^3}=-2 when \dfrac{x^2+1}{x}=1

Solution:

\dfrac{x^2+1}{x}=\dfrac{x^2}{x}+\dfrac{1}{x}=x+\dfrac{1}{x}

\therefore from the given condition we have,

x+\dfrac{1}{x}=1 \\    or, \: x^3+\dfrac{1}{x^3}=(x+\dfrac{1}{x})^3-3.x.\dfrac{!}{x}.(x+\dfrac{1}{x}) \\    or, \: 1^3-3.1=1-3=-2 (Proved)

Formula:

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

Again, (a-b)^3=a^3-b^3-3ab(a-b) [Taking -3ab common from -3a^2b+3ab^2 ]

Proof:

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

Corollary: a^3-b^3=[a^3-b^3-3ab(a-b)]+3ab(a-b)

or, a^3-b^3=(a-b)^3+3ab(a-b)

Example: Find the cube of 3x-4y

Solution:

(3x-4y)^3=(3x)^3-3(3x)^2(4y)+3(3x)(4y)^2-(4y)^3 \\    =27x^3-3(9x^2)(4y)+3(3x)(16y^2)-64y^3 \\    =27x^3-108x^2y+144xy^2-64y^3

Example: Find the cube of (a-b-c)

Solution:

(a-b-c)^3=[(a-b)-c]^3=(a-b)^3-3(a-b)^2c+3(a-b)c^2-c^3 \\    =(a^3-3a^2b+3ab^2-b^3)-3(a^2+b^2-2ab)c+3(a-b)c^2-c^3 \\    =a^3-b^3-c^3-3a^2b+3ab^2-3a^2c+3ac^2-3b^2c-3bc^2+6abc

Example: Find the cube of 297

Solution:

297^3=(300-3)^3=300^3-3^3-3.300.3.(300-3) [\because (a-b)^3=a^3-b^3-3ab(a-b) ] \\    =27000000-27-810000+8100 \\    =(27000000+8100)-(27+810000) \\    =27008100-810027

Example: If \dfrac{x^2-1}{x}=4 , find the value of \dfrac{x^6-1}{x^3}

Solution:

\dfrac{x^2-1}{x}=\dfrac{x^2}{x}-\dfrac{1}{x}=x-\dfrac{1}{x}

\therefore \dfrac{x^2-1}{x}=4 …(1)

[\because \dfrac{x^2-1}{x}=4 ]

\dfrac{x^6-1}{x^3}=\dfrac{x^6}{x^3}-\dfrac{1}{x^3}=x^3-\dfrac{1}{x^3} \\    =(x-\dfrac{1}{x})^3+3.x.\dfrac{1}{x}.(x-\dfrac{1}{x})

=4^3+3.4=64+12=76. [Using (1)]

Exercise 1:

1. Find the cube of:

a) \dfrac{2}{3a}+\dfrac{3}{5b}

b) xy+yz

c) 2x-y-z

d) p^2-q

2. Simplify:

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

b) (5x-8)^3-(3x-8)^3-6x(5x-8)(3x-8)

c) (a-b+c)^3+(a+b-c)^3+6a[a^2-(b-c)^2]

d) (5x-2)^3+(3-4x)^3+3(x+1)(5x-2)(3-4x)

3. Find the value of a^3+b^3 when a+b=6 and ab=7

4. Find the value of a^3-b^3 when a-b=4 and ab=2

5. If a+\dfrac{1}{a}=3 , show that, a^3+(\dfrac{1}{a})^3=18

6. If \dfrac{a^2+1}{a}=4 show that \dfrac{a^6+1}{a^3}=52

7. If x-\dfrac{1}{x}=p show that x^3-\dfrac{1}{x^3}=p^3+3p

8. If \dfrac{a^2-1}{a}=1 find the value of \dfrac{a^6+1}{a^3}

Formula:

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

Proof:

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

Conversely, a^3+b^3=(a+b)(a^2-ab+b^2) . Hence, any expression of the form a^3+b^3 can be resolved into factors.

Example: Multiply x^4-x^2+1 by x^2+1

Solution:

Putting a for x^2 and b for 1 we have,

x^4-x^2+1=(x^2)^2-x^2.1+1^2=a^2-ab+b^2

Hence, (x^2+1)(x^4-x^2+1)=(a+b)(a^2-ab+b^2) \\    =a^3+b^3 \\    =(x^2)^3+1^3=x^6+1

Example: Resolve into factors a^3b^3+8c^3

Solution:

a^3b^3+8c^3=(ab)^3+(2c)^3 \\    =(ab+2c)[(ab)^2-(ab)(2c)+(2c)^2] \\    =(ab+2c)(a^2b^2-2abc+4c^2)

Example: Resolve into factors 64p^3+125

Solution:

64p^3+125=(4p)^3+(5)^3 \\    =(4p+5)[(4p)^2-(4p)(5)+(5)^2] \\    =(4p+5)(16p^2-20p+25)

Formula:

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

Proof:

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

Conversely, a^3-b^3=(a-b)(a^2+ab+b^2) . Hence, any expression of the form a^3-b^3 can be resolved into factors.

Example: Multiply 16a^2+20ab+25b^2 by 4a-5b

Solution:

Putting x for 4a and y for 5b we have,

16a^2+20ab+25b^2=(4a)^2+(4a)(5b)+(5b)^2 \\    =x^2+xy+y^2

\therefore (4a-5b)( 16a^2+20ab+25b^2) \\    =(x-y)(x^2+xy+y^2) \\    =x^3-y^3 \\    =(4a)^3-(5b)^3 \\    =64a^3-125b^3

Example: Resolve into factors 64x^3-a^3y^6

Solution:

64x^3-a^3y^6=(4x^2)^3-(ay^2)^3 \\    =(4x^2-1)[(4x^2)^2+(4x^2)(ay^2)+(ay^2)^2] \\    =(4x^2-ay^2)(16x^4+4ax^2y^2+a^2y^4)

Example: Resolve into factors 343x^3-8y^6

Solution:

343x^3-8y^6=(7x)^3-(2y^2)^3 \\    =(7x-2y^2)[(7x)^2+(7x)(2y^2)+(2y^2)^2] \\    =(7x-2y^2)(49x^2+14xy^2+4y^4)

Exercise 2:

1. Multiply:

a) x^2-x+1 by x+1

b) 1-2x+4x^2 by 1+2x

c) 1+2x+4x^2 by 1-2x

d) x^2+3x+9 by x-3

2. Resolve into factors:

a) 1331a^3b^6x^9+729c^3y^6z^9

b) 27a^3+343y^3

c) 125a^2 -1

d) 1-512k^3

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  • Complex Numbers
  • Quadratic Equations
  • Logarithm
  • Permutation
  • Combination
  • More on Complex Numbers
  • Classification of Numbers
  • Positive and Negative Quantities
  • Understanding Simple Algebraic Formulas With Examples
  • Integers
  • Linear Inequalities
  • An Introduction to Fundamental Algebra
  • Basic Number Properties – Commutative, Associative and Distributive
  • Algebraic Multiplication and Division
  • Simple Equations in One Variable
  • Simple Formulae and their Application
  • Rational and Irrational Numbers
  • Problems Leading to Simple Equations
  • Simultaneous Equations
  • Mathematical Induction
  • Different Type of Sets
  • Indices
  • Framing Formulas
  • Sequences
  • Introduction to Matrices
  • Addition Of Matrices
  • Subtraction Of Matrix
  • Multiplication of Matrices
  • Determinant of Matrices
  • Co-factor of Matrices
  • Minor of Matrices
  • Transpose and Adjoint of Matrices
  • Inverse of a Matrix
  • System of Linear Equations in Matrices
  • Introduction to Polynomials
  • Classification of Polynomials
  • Addition and Subtraction of Polynomials
  • Multiplication of Polynomials
  • Factoring Polynomials
  • Zeroes of Polynomial
  • Remainder Theorem of Polynomials
  • Factor Theorem of Polynomial
  • Simplifying Polynomial Fractions
  • Roots of a Polynomial
  • Addition of Polynomial Fractions
  • Subtraction of Polynomial Fractions
  • Multiplying polynomial fractions
  • Division of Polynomial Fractions

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