Review of Basic Algebra

(A Review for SSI and Challenge 2000)

I. Exponents

If n is a positive integer, then aⁿ represents the product of n factors each of which is a.

In general,

Example: a x a x a= a³ and 2 x 2 x 2 x 2 = 2⁴ = 16

Definition: Given the expression , where a, n, and b are real numbers (a, n, and bR ), a is the base and n is the exponent or power.

Example: In 5² , 5 is the base and 2 is the exponent.

Note: If n is a positive integer and if a and b are real numbers such that aⁿ = b, then a is said to be an n^th root of b and may be written

( or ).

For Negative Integral Exponents:

If n is a positive integer, .

Example: 3^-2 =

Positive Fractional Exponents:

, where m and n are positive integers.

Example:

Negative Fractional Exponent:

Example:

Zero Exponent:

a ⁰ = 1, a 0

Example: b ⁰ = 1 , 5 ⁰ = 1, and 333 ⁰ = 1

General Laws of Exponents:

a ^p a ^q = = a ^{p + q}

Example: 5² x 5³ = (5) (5) x (5) (5) (5) = 5 ⁵ = 3125

( a ^p )^q = = a ^pq

Example: ( 3 ² )³ = ( 3² ) ( 3² ) ( 3² ) = 3 ^{(2) (3)} = 3 ⁶ = 729

= a ^p = a ^p a^-q = a ^p-q , a 0

Recall: a^-q =

Example:

( ab ) ^p = = a ^p b ^p

Example: (2 x 3)² = (2)(3) x (2)(3) = 2² x 3² = 4 x 9 = 36

( a / b) ^p , = , b 0

Example:

Some Important Formulas (only a very few are to be memorized!)

A) (a+ b) (a - b) = (a )( a) - (a)( b) + (a)( b) - (b)( b) = a² - ab + ab - b² = a² - b²

Example: (3a + b)(3a - b) = (3a)( 3a)-(3a)( b) + (3a)(b) - (b)(b) = 9a² -3ab+ ab - b² = 9a² - b²

(a + b) ( a - b) = a² - b²

B) (a + b)(a + b) = (a + b)² = (a)( a) + (a )( b) + (a )( b) + (b )( b) = a² + ab + ab + b² = a² + 2ab + b²

Example: (2a + b)(2a + b) = (2a + b)² = (2a)( 2a) + (2a)( b) + (2a)( b) + (b)( b) = 4a² + 2ab + 2ab + b² = 4a² + 4ab + b²

(a + b) ( a + b) = a² + 2ab + b²

C) (a -b)(a - b) = (a -b)² = (a)( a) - (a)(b ) - (a)( b) + (b)(b) = a² - ab -ab + b² = a² -2ab + b²

Example: (2a -b) ( 2a -b) = ( 2a - b)² = (2a)( 2a) - (2a )(b) - (2a)(b) + (b)(b) = 4a² - 2ab - 2ab + b² = 4a² - 4ab + b²

(a - b)(a - b) = a² - 2ab + b²

D) (x + a )( x + b ) = (x )( x) + ( b)( x) + ( a )( x) +( a )( b) = x² + bx + ax + ab = x² + (a + b ) x + ab

Example: (3x + 2a) (3x + 2b) = (3x)(3x) + ( 2b)( 3x) + (2a)(3x) + (2a)( 2b)= 9x² + 6bx + 6ax + 4ab

(TO MEMORIZE OR KNOW!): (x + a)(x + b) = x² + (a + b)x + ab

III. Quadratic Formula

Consider the polynomial: ax² + bx + c.

ax² + bx + c = 0 , where a 0.

The solutions of this quadratic equation are the values of x for which the above polynomial is zero. They are also called roots or zeroes of the plynomial ax² + bx + c. These root can be expressed in terms of the coefficients a, b, and c as shown below.

Given , the discriminant or

x = .

A) If the discriminant is equal to zero (b² - 4ac = 0) the roots are real and equal.

Example: Given 4x² - 12x + 9 = 0 a = 4 , b = -12 and c = 9

Therefore, x =

Since the discrinimant is 0, the roots are equal.

Note: A rational number is one that can be expressed in the fractional form , where and b are integers and . It can further be defined as the ratio of two integers.

Example: 5 = 5/1 , 1/2 , 3/4 and 4/5 are rational numbers. and are irrational because they cannot be written as the ratio of two integers.

B) If the discriminant is less than zero (b² - 4ac < 0), the roots are complex numbers.

Example: Given x² + 3x + 5 = 0 a = 1 , b = 3 and c = 5

Therefore, x =

x_{1 = and x2 = . Note well that}

The above roots are complex numbers, i.e., any number written as Both are real numbers.

Example: is a pure imaginary number.

A complex number may be defined as a real number plus an imaginary number and may be written in the form z = a + ib.

C) If the discriminant is greater than zero ( b² - 4ac > 0 ) then the roots are real and unequal.

Example: Given 5x² + 2x - 9 = 0 a = 5 , b = 2 and c = - 9

Therefore, x =

x₁ = and x₂ =

Since the discriminant is greater than zero, the roots and unequal.