Fundamentals Of Algebra

FUNDAMENTALS OF ALGEBRA POLYNOMIALS A variable is a letter that can represent any number from a given set of real numbers. The common variables used are x, y, z etc. When we use addition, subtraction, multiplication, division, powers, roots on the variables, we get algebraic expressions. w w w w w w w

Examples : 2x 2 @ 3x + 2,

2 p y + x2 ,

xf +f yf f f f f f f f f f f f f f f etc are algebraic expressions. x @y

A monomial is a real number or a real number multiplied with variables with powers as 2f f3 2 whole numbers. 6x 2 , 2y, @ 5, @ f x y z are examples of monomials. 3

Sum of two or more monomials is called a polynomial. [Sum of two monomials is called a binomial.] 1f 1f f2 f2 Example: f x @ 3x + 2 is a polynomial which has three terms f x , @ 3x and 2 A 3 3 COMBINING ALGEBRAIC EXPRESSIONS: Addition and Subtraction :

To combine polynomials, we use the same properties of real number operations (see the chapter : Sets, Numbers, Operations, Properties. The terms in which the same variables are raised to the same powers are the like terms. 1f f 3 xy z, @ x 2 , 3xy 3 z, 10xy, 7xy 2 the like terms are Among the terms 3x 2 , @ 4xy 2 , 2xy, @ f 2 1f f 3 3x 2 and @ x 2 , @ 4xy 2 and 7xy 2 , 2xy and 10xy, @ f xy z and 3xy 3 z A 2 To add or subtract two polynomials, we combine the like terms by using the Distributive Property. Example : 3x 2 + 5x 2 = 3 + 5 x 2 = 8x 2 `

a

If there is a minus sign before a parentheses, while removing the parentheses, signs of all thebterms inside cthe parentheses are changed. @ 2x 2 @ 3x + 4 = @ 2x 2 + 3x @ 4

Example : (a) Find the sum of 2x 3 @ 3x 2 + x @ 5 and x 3 + 2x 2 @ 2x + 1 b

c

b

(b) Subtract x 3 + 2x 2 @ 2x + 1 from 2x 3 @ 3x 2 + x @ 5 b

c

b

Solution : b c b c ` a a 2x 3 @ 3x 2 + x @ 5 + x 3 + 2x 2 @ 2x + 1 b

c b

c `

c

c

= 2x 3 + x 3 + @ 3x 2 + 2x 2 + x @ 2x + @ 5 + 1 a `

a

= 3x 3 @ x 2 @ x @ 4 c b c b 2x 3 @ 3x 2 + x @ 5 @ x 3 + 2x 2 @ 2x + 1

` a b

= 2x 3 @ 3x 2 + x @ 5 @ x 3 @ 2x 2 + 2x @ 1 b c b c ` a ` a = 2x 3 @ x 3 + @ 3x 2 @ 2x 2 + x + 2x + @ 5 @ 1 = x 3 @ 5x 2 + 3x @ 6

Multiplication :

To find the product of polynomials we use the Distributive property and laws of exponents. Example : Find the product 3x 2 @ 5 x 3 + 2x 2 @ 2x + 1 b

cb

Solutionb: cb c 3x 2 @ 5 x 3 + 2x 2 @ 2x + 1

c

= 3x 2 x 3 + 2x 2 @ 2x + 1 @ 5 x 3 + 2x 2 @ 2x + 1 b

c

b

c

Distributive property

= 3x 5 + 6x 4 @ 6x 3 + 3x 2 @ 5x 3 @ 10x 2 + 10x @ 5 Distributive property = 3x 5 + 6x 4 @ 11x 3 @ 7x 2 + 10x @ 5 Combining the like terms

SOME SPECIAL PRODUCT FORMULAS: 1A A + B A @B = A @B `

a`

2

a

2 A A + B = A + 2AB + B a2

`

2

2

3 A A @ B = A @ 2AB + B a2

`

2

2

2

4 A A + B = A + 3A B + 3AB + B a3

`

3

2

2

3

5 A A @ B = A @ 3A B + 3AB @ B a3

`

3

ab

2

2

c

6 A A + B A @ AB + B = A + B `

ab

2

2

3

c

3

7 A A @ B A + AB + B = A @ B `

2

2

3

3

3

Example : Find 2x @ 3 and 2 + 3y 2 @ 3y a2

`

b

cb

c

Solution : `

a2

2x @ 3

= 2x @ 2.2x A 3 + 3 a2

`

2

taking A = 2x,B = 3 and using A @ B = A @ 2AB + B

2

taking A = 2,B = 3y and using A + B A @ B = A @ B

2

`

= 4x 2 @ 12x + 9

a2

2

2 + 3y 2 @ 3y

b

cb

c2

= 2 @ 3y 2

b

c `

= 4 @ 9 y2

a`

a

2

FACTORING Factoring is the opposite process of finding products. Here a polynomial is to be written as the product of two or more simpler polynomials. If` we multiply 2xa @ 3 `with 4xa + 5`, we geta a` 2x @ 3 4x + 5 = 2x 4x + 5 @ 3 4x + 5

= 8x 2 + 10x @ 12x @ 15 = 8x 2 @ 2x @ 15 On the reverse, if` we area`asked to factor 8x 2 @ 2x @ 15, we have to write a 2 8x @ 2x @ 15 = 2x @ 3 4x + 5 . ` a ` a Here, 2x @ 3 and 4x + 5 are the factors of 8x 2 @ 2x @ 15.

To factor a polynomial, we use the following methods wherever applicable :

Factoring out common factors:

If all the terms of a polynomial have one or more common factors, we take out the greatest common factor and use the Distributive Property. Example : Factor each polynomial/expression : ` a 2 a 5x y @ 10xy

b 6ab c 3 @ 18a 2 c 2 + 12ac 2 ` a` a` a ` a c 2x + 4 2x @ 4 @ 3 2x @ 4 2

` a

Solution : ` a 2 a 5x y = 5 Bx Bx By 10xy = 5 B2 Bx By

The greatest common factor is 5 Bx By or 5xy

5x 2 y @ 10xy = 5xy Bx @ 5xy B2 ` a = 5xy x @ 2

Hence

b 6ab c 3 = 2 B3 Ba Bb Bb Bc Bc Bc

` a

2

18a 2 c 2 = 2 B3 B3 Ba Ba Bc Bc 12ac 2 = 2 B2 B3 Ba Bc Bc The geatest common factor is 2 B3 Ba Bc Bc or 6ac 2 Hence b c 2 2 6ab c 3 @ 18a 2 c 2 + 12ac 2 = 6ac 2 b c @ 3a + 2 c 2x + 4 2x @ 4 @ 3 2x @ 4 ` a the greatest common factor is 2x @ 4

` a`

Hence,

a`

a

`

a

2x + 4 2x @ 4 @ 3 2x @ 4 = 2x @ 4

`

a`

a

`

a `

2x + 4 @ 3

aB`

= 2x @ 4 2x + 1 `

a`

a

a

C

Factoring by using the Product Formulas given above:

See the right hand sides of all the special product formulas given above. The left hand sides are in the factor form. If any expression is in the form of the right hand sides, we can use the reverse of that. Example` :aFactor a 4x 2 @ 49

b 27x 3 + c 3

` a

c 4a 3 @ 32b

` a

3

Solution : ` a 2 ` a2 2 a 4x @ 49 = 2x @ 7 ` a` a = 2x + 7 2x @ 7 b 27x 3 + c 3 = 3x + c 3

` a

`

a3

using the reverse of A + B A @ B = A @ B `

a2

aB`

= 3x + c 9x 2 @ 3cx + c 2 ab

`

c 4a 3 @ 32b = 4 a 3 @ 8b

` a

3

b

c

3

aB

c

`

C

using the reverse of a2C

= 4 a @ 2b a 2 + a A 2b + 2b `

`

= 4 a @ 2b a 2 + 2ab + 4b `

2

a

2

using the reverse of A + B A @ AB + B = A + B

= 3x + c 3x @ 3x A c + c 2 `

a`

ab

c

2

`

ab

c

2

2

3

A @ B A + AB + B = A @ B ab

2

c

2

3

3

3

Factoring by grouping :

Example` : aFactor a 2x 3 + x 2 @ 6x @ 3

b @ 6x 3 @ 2x 2 + 3x + 1

` a

Solution : b c ` ` a 3 a a 2x + x 2 @ 6x @ 3 = 2x 3 + x 2 @ 6x + 3

Group terms

= x 2 2x + 1 @ 2 2x + 1 `

a

`

= x 2 @ 2 2x + 1 b

c`

a

a

factor out common factors

b @ 6x 3 @ 2x 2 + 3x + 1 = @ 6x 3 + 2x 2 + 3x + 1

` a

b

c `

a

= @ 2x 2 3x + 1 + 1 3x + 1 `

a

`

= @ 2x 2 + 1 3x + 1 b

c`

a

a

RATIONAL EXPRSSIONS A rational expression is an expression of the type ` a

` a

Pf xf f f f f f f f f f f f f f f ` a where Q x

` a

` a

P x andQ x are

Pf xf f f f f f f f f f f f f f ` a is to be in the simplest form (i.e. after canceling polynomials and Q x ≠ 0. Also, f Q x out any common factor other than 1 or -1). ` a

2 1f @ xf @ 2f + 1f xf f f f f f f f f f f f f f f 3x f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f b c are rational expressions. Examples : f , f , f 3 x @1 x @2 x x @1

Some important operations of Rational Expressions: b c These are similar to operations with fractions If A, B, C, D are polynomials, then AC f f f f f f f f f f f A f f f f f = 1A BC B Af f f f fC f f f f f f AC f f f f f f f f f f f 2A A = B D BD Af f f f f C f f f f f f AD f f f f f f f f f f f 3A D = B D BC Af +f B f f f f B f f f f f A f f f f f f f f f f f f f f f f f f 4A + = C C C Af @ B f f f f B f f f f f A f f f f f f f f f f f f f f f f f f f 5A @ = C C C Af f f f f C f f f f f f 6 A = if and only if AD = BC B D

Example : Simplify 2 ` a xf @ xf @ 2f f f f f f f f f f f f f f f f f f f f f f f f f f f f f a f 2 x @1 Solution :

` a f xf 3f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f

b

x+1

@

x @2

` a` a xf @ 2f xf +f 1 f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f a` a =`

2 ` a xf @ xf @ 2f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f

a

x @1 x + 1 xf @ 2f f f f f f f f f f f f f f f = f x @1 ` a ` a 3f xf @ 2f xf xf +f 1f ` a f xf 3f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f a` a@` a` a b @ =` x @2 x + 1 x + 1 x @2 x + 1 x @2 ` a ` a 3f xf @ 2f @ xf xf + 1f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f a` a = ` x @2 x + 1 x2@ 1

2 @ xf 3x @ 6f @ xf f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f ` a` a = f x @2 x + 1 2 @ xf + 2x @ 6f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f a` a = `f x @2 x + 1

Example: Simplify 2 ` a xf 2f 3f @ 4f @ 3x @ 4f 1f f f f f f f f f f f f f f f f f f xf f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f ` a f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f a 2 D 2 b @` + 2 a 2 x+1 x @ 1 x @ 4 x + 5x + 6 x+1 Solution: 2 ` a xf @ 4f @ 3x @ 4f f f f f f f f f f f f f f f f f xf f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f D a f 2 2 x @ 4 x + 5x + 6 2 xf @ 4f +f 5x + 6f f f f f f f f f f f f f f f f f xf f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f = f B 2 2 4 x @` 4 xa`@ 3x @ a` a x @ 4 x + 2 xf +f 3f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f a` a` a` a =` x + 2 x @2 x @4 x + 1 xf +f 3f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f a` a = `f x @2 x + 1 ` a f 2f 3f 1f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f b f @` f + f a 2 2 x+1 x+1 x @1

1f 2f 3f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f a` a+ ` a` a = f @ `f x+1 x+1 x+1 x + 1 x @1 ` a` a xf +f 1f xf @ 1f 1f 2f @ 1f 3f +f 1f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f xf f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f xf f f f f f f f f f f f f f a` a@ ` a` aB a` aB = B` + `f x + 1 x + 1 x @1 x + 1 x + 1 x @1 x + 1 x @1 x + 1 ` a` a ` a ` a xf +f 1f xf @ 1f @ 2f xf @ 1f +f 3f xf +f 1f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f ` a` a` a = x @1 x + 1 x + 1 2 xf @ 1f @ 2x + 2f +f 3x + 3f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f ` a` a` a = f x @1 x + 1 x + 1 2 xf +f xf + 4f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f a` a` a = `f x @1 x + 1 x + 1

Some reminders in algebra: 1A 2A

`

a + b ≠ a2 + b a2

w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w qa 2 + b 2 ≠ a +

2

b

as

`

a + b = a + b a + b = a 2 + 2ab + b a2

`

as squaring both sides we get

a`

a

a2 + b ≠ a + b 2

`

2

a2

3A

1f 1f f f f 1f f f f f f f f f f f f f f f f f f f + ≠ a b a+b

as

bf af 1f + af f f f 1f f f f 1f f f f bf f f f 1f f f f af f f f f f f f f f f f f f f f f f bf f f f f f f f f f f f f f f f + = B + B = + = a b a b b a ab ab ab

4A

af +f bf f f f f f f f f f f f f f ≠b a

as

af +f bf bf f f f f f f f f f f f f f af f f f bf f f f f f f = + =1+ a a a a

as

w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w q a 2 A b 2 = q` a A ba2 = ab

w w w w w w w w w w w w w w w w w w w w w w w w w w 2 q 2 5A a Ab = aAb

f

but

ab f f f f f f f f =a b

g