Checking Factoring  The checking of factoring can be done with the calculator.  Graph the following expressions: 1.x 2 + 5x – 6 2.(x – 3)(x – 2) 3.(x + 6)(x – 1)  What do you notice?  Are they the same graph?  Discuss what you can conclude from the graphs.

Roots  What is the value of 2 2 ?  What is x?  However: x =  In this case 2 is considered the principal root or the nonnegative root, when there is more than one real root.  Finding the square root of a number and squaring a number are inverse operations. WHY? Should be 2, right?

Roots  What is the value of  The values found are known as the nth roots and are also principal roots.  The following is the format for a radical expression. index radical sign radicand

Roots of  Summary of the real nth roots. Real nth Roots of b,, or – nb > 0b < 0b = 0 even one positive root one negative root no real roots one real root, 0 odd one positive root no negative roots no positive roots one negative root

Practice with Roots  Simplify the following  has to be absolute value to identify principal root  Estimated between 5 and 6 because 5 2 = 25 and 6 2 = 36.

Radical Expressions  Radical “like expressions” have the same index and same radicand.  Product and Quotient Properties:

Simplifying Radical Expressions 1.The index, n, has to be as small as possible 2.radicand  NO factors, nth roots 3.radicand  NO fractions 4.NO radical expressions in denominator  For example:

 More examples: Simplifying Radical Expressions

 More examples: Simplifying Radical Expressions

Radical Expressions  Conjugates  ± same terms  Multiply the following:

Radical Expressions  Deduction about conjugates:  Product of conjugates is always a rational number.  For example:

Radical Expressions  Simplify the following

Radical Expressions  In-Class work  Rationalize the denominator: