1. Unit 5 Seminar Agenda Exponents Scientific Notation Polynomials and Polynomial Functions Multiplying Polynomials

2. Laws of Exponents PRODUCT RULE OF EXPONENTS. (a x ) * (a y ) = a (x + y) (KEEP THE BASE and ADD THE EXPONENTS.) Example: (6 5 )(6 8 ) = 6 5+8 = 6 13 Example: a 8 a 12 QUOTIENT RULE OF EXPONENTS. (a x ) / (a y ) = a (x - y) (KEEP THE BASE and SUBTRACT THE EXPONENTS) Example:

3. Anything to the zero power is 1. a 0 =1 Anything to the first power is itself. a 1 =a POWER RULE OF EXPONENTS. (a x ) y = a xy ( Keep The Base and MULTIPLY THE EXPONENTS.) Example: (w 5 ) 8 = w 58 = w 40 Example: (b 4 ) 3

4. A negative exponent moves the term to the other side of the fraction bar. a -1 = 1/a and 1/a -1 = a Example: The next law of exponents is a combination of the product rule and the power rule. (ax 3 ) 2 = a 2 * x 2 * 3 = a 2 x 6 Example: The final law of exponents is a combination of the quotient rule and the power rule. (x/y) 3 = x 3 /y 3 Example:

5. Example Example: simplify the expression: Do not be alarmed!! Raise each term to the –3 power then we’ll go from there. I’m going to go ahead and multiply powers as I go.

6. The next thing I recommend is that you move all terms with negative exponents such that the exponents are positive. I call this process the “move it or leave it” game. Here’s how the game works. If it has a negative exponent, move it to the other side of the fraction bar, making the exponent positive. If it has a positive exponent already, leave it alone! Almost there! Calculate 4 3 and 3 3 since you can, and alphabetize the variables (traditionally done).

7. Scientific Notation 1) Change.000000345 to Scientific Notation: Move the decimal 7 places to the right 3.45 x 10 -7 2) Change 567,000,000,000 to Scientific Notation: Move the decimal 11 places to the left 5.67 x 10 11

8. Exampes: 2.793 x 10 9 0.00000571 =

9. Polynomials One term – it’s a MONOMIAL Two terms – it’s a BINOMIAL Three terms – it’s a TRINOMIAL Four or more terms – it’s a POLYNOMIAL, sometimes clarified by naming exactly how many terms there are; ex. a polynomial with six terms Certain situations cause an expression to NOT be a polynomial. There’s a variable in a denominator There’s a variable under a radical There’s a number or variable with a fractional or negative exponent

10. Examples of Adding and Subtracting Polynomials: (x + 4)(x – 3) (4x – 5) (3x + 7) (x 2 +6x-3)(3x 2 -4x+2)

11. 14 x 23 You would multiply 3*4, then 3*1 to get your first partial product. 1 14 x 23 42 Then you would place a zero as a placeholder to make sure that your next partial product starts in the tens column. Multiply 2*4 and 2*1 to get the second partial product. 1 14 x 23 42 280 Finally, you would add the partial products to reach the final product. 1 14 x 23 142 + 280 322

12. Vertical Method for multiplying 2 binomials Alright, let’s return to the polynomial problem. Keep in mind what order you followed to multiply this problem. x + 4 x – 3 Starting with the “ones” column, multiply -3*4 and -3*x to get the first partial product. By the way, there is no “carrying” if the products are double digit. x + 4 x – 3 -3x - 12 Moving to the “tens” column, multiply x*4 and x*x to get the second partial product. Line up your like terms as you go. x + 4 x – 3 -3x - 12 x 2 + 4x Finally, add the two partial products together. In other words, COMBINE LIKE TERMS. x + 4 x – 3 -3x - 12 x 2 + 4x____ x 2 + x - 12

13. (4x – 5) (3x + 7) 4x – 5 3x + 7 28x – 35 12x 2 – 15x_____ 12x 2 + 13x – 35

14. (x 2 +6x-3)(3x 2 -4x+2) x 2 + 6x – 3 3x 2 – 4x + 2 2x 2 + 12x – 6 -4x 3 – 24x 2 + 12x 3x 4 + 18x 3 – 9x 2 __________ 3x 4 + 14x 3 – 31x 2 + 24x – 6

15. Horizontal Method (FOIL) for multiplying 2 binomials (x + 3)(x + 5) Round one: distribute the first term in the first quantity (set of parentheses) to the second entire quantity. x(x + 5) = x 2 + 5x Round two: distribute the second term (including the sign) in the first quantity to the second entire quantity. 3(x + 5) = 3x + 15 Round three: combine like terms. x 2 + 5x + 3x + 15 = x 2 + 8x + 15 The product (answer) is a trinomial (three-termed polynomial). This is often the case.

16. Examples: (x + 6)(x – 3) (3x – 7) 2

17. Special Cases (x+a)(x-a) = x 2 – a 2 (difference of 2 squares) (x+a) 2 = (x+a)(x+a) = x 2 + 2ax + a 2 (perfect square trinomial)

18. Choose the correct answer: 1) Simplify: (x 3 y -7 )/(xy 2 ) = a. x 2 /y 9 b. x 4 /y 5 2)Multiply: (x+4)(2x-3) = a. 2x 2 +3x-12 b. 2x 2 +5x-12 3)Multiply: (x-2)(x 2 +2x+4) = a. x 3 -8 b. x 3 +4x-8

19. Questions?