5.1&5.2 Exponents

1. 5.1&5.2 Exponents 82 =8 • 8 = 64 24 = 2 • 2 • 2 • 2 = 16 x2 = x • x x4 = x • x • x • x Base = x Base = x Exponent = 2 Exponent = 4 Exponents of 1 Zero Exponents Anything to the 1 power is itself Anything to the zero power = 1 51 = 5 x1 = x (xy)1 = xy 50 = 1 x0 = 1 (xy)0 = 1 Negative Exponents 5-2 = 1/(52) = 1/25 x-2 = 1/(x2) xy-3 = x/(y3) (xy)-3 = 1/(xy)3 = 1/(x3y3) a-n = 1/an 1/a-n = an a-n/a-m = am/an

2. Powers with Base 10 100 = 1 101 = 10 102 = 100 103 = 1000 104 = 10000 100 = 1 10-1 = 1/101 = 1/10 = .1 10-2 = 1/102 = 1/100 = .01 10-3 = 1/103 = 1/1000 = .001 10-4 = 1/104 = 1/10000 = .0001 The exponent is the same as the The exponent is the same as the number number of 0’s after the 1. of digits after the decimal where 1 is placed Scientific Notation uses the concept of powers with base 10. Scientific Notation is of the form: __. ______ x 10 (** Note: Only 1 digit to the left of the decimal) You can change standard numbers to scientific notation You can change scientific notation numbers to standard numbers

3. Scientific Notation Scientific Notation uses the concept of powers with base 10. Scientific Notation is of the form: __. ______ x 10 (** Note: Only 1 digit to the left of the decimal) -2 5 321 Changing a number from scientific notation to standard form Step 1: Write the number down without the 10n part. Step 2: Find the decimal point Step 3: Move the decimal point n places in the ‘number-line’ direction of the sign of the exponent. Step 4: Fillin any ‘empty moving spaces’ with 0. Changing a number from standard form to scientific notation Step1: Locate the decimal point. Step 2: Move the decimal point so there is 1 digit to the left of the decimal. Step 3: Write new number adding a x 10n where n is the # of digits moved left adding a x10-n where n is the #digits moved right 5.321 .05321 .0 5 3 2 1 = 5.321 x 10-2

4. Raising Quotients to Powers a n b an bn a -n b a-n b-n bn an b n a = = = = Examples: 3 2 32 9 4 42 16 = = 2x 3 (2x)3 8x3 y y3 y3 = = 2x -3 (2x)-3 1 y3 y3 y y-3 y-3(2x)3 (2x)3 8x3 = = = =

5. Product Rule am• an = a(m+n) x3• x5 = xxx • xxxxx = x8 x-3 • x5 = xxxxx = x2 = x2 xxx 1 x4 y3 x-3 y6 = xxxx•yyy•yyyyyy = xy9 xxx 3x2 y4 x-5• 7x = 3xxyyyy • 7x = 21x-2 y4 = 21y4 xxxxx x2

6. Quotient Rule am = a(m-n) an 43= 4 • 4 • 4 = 41 = 4 43 = 64 = 8 = 4 42 4 • 4 42 16 2 x5 = xxxxx = x3x5 = x(5-2) = x3 x2 xx x2 15x2y3 = 15 xx yyy = 3y215x2y3 = 3 • x -2 • y2 = 3y2 5x4y 5 xxxx y x2 5x4y x2 3a-2 b5 = 3 bbbbb bbb = b83a-2 b5 = a(-2-4)b(5-(-3)) = a-6 b8 = b8 9a4b-3 9aaaa aa 3a6 9a4b-3 3 3 3a6

7. Powers to Powers (am)n = amn (a2)3 a2• a2 • a2 = aa aa aa = a6 (24)-2 = 1 = 1 = 1 = 1/256 (24)2 24• 24 16 • 16 28 256 (x3)-2 = x –6 = x 10 = x4 (x -5)2 x –10 x 6 (24)-2 = 2-8 = 1 = 1

8. Products to Powers (ab)n = anbn (6y)2 = 62y2 = 36y2 (2a2b-3)2 = 22a4b-6 = 4a4 = a 4(ab3)3 4a3b9 4a3b9b6 b15 What about this problem? 5.2 x 1014 = 5.2/3.8 x 109 1.37 x 109 3.8 x 105 Do you know how to do exponents on the calculator?

9. Square Roots & Cube Roots A number b is a cube root of a number a if b3 = a 8 = 2 since 23 = 8 Notice that 8 breaks down into 2 • 2 • 2 So, 8 =  2 • 2 • 2 See a ‘group of 3’ –> bring it outside the radical (the cube root sign) Example: 200 = 2 • 100 = 2 • 10 • 10 = 2 • 5 • 2 • 5 • 2 = 2 • 2 • 2 • 5 • 5 = 2 25 A number b is a square root of a number a if b2 = a 25 = 5 since 52 = 25 Notice that 25 breaks down into 5 • 5 So, 25 =  5 • 5 See a ‘group of 2’ -> bring it outside the radical (square root sign). Example: 200 = 2 • 100 = 2 • 10 • 10 = 10 2 3 3 3 3 3 3 3 3 Note: -25 is not a real number since no number multiplied by itself will be negative Note: -8 IS a real number (-2) since -2 • -2 • -2 = -8 3

10. 5.3 Polynomials • TERM • a number: 5 • a variable X • a product of numbers and variables raised to powers 5x2 y3 p x(-1/2)y-2 z MONOMIAL -- Terms in which the variables have only nonnegative integer exponents. -4 5y x2 5x2z6 -xy7 6xy3 A coefficient is the numeric constant in a monomial. POLYNOMIAL - A Monomial or a Sum of Monomials: 4x2 + 5xy – y2(3 Terms) Binomial – A polynomial with 2 Terms (X + 5) Trinomial – A polynomial with 3 Terms DEGREE of a Monomial – The sum of the exponents of the variables. A constant term has a degree of 0 (unless the term is 0, then degree is undefined). DEGREE of a Polynomial is the highest monomial degree of the polynomial.

11. Adding & Subtracting Polynomials Combine Like Terms (2x2 –3x +7) + (3x2 + 4x – 2) = 5x2 + x + 5 (5x2 –6x + 1) – (-5x2 + 3x – 5) = (5x2 –6x + 1) + (5x2 - 3x + 5) = 10x2 – 9x + 6 Types of Polynomials f(x) = 3 Degree 0 Constant Function f(x) = 5x –3 Degree 1 Linear f(x) = x2 –2x –1 Degree 2 Quadratic f(x) = 3x3 + 2x2 – 6 Degree 3 Cubic

12. 5.4 Multiplication of Polynomials Step 1: Using the distributive property, multiply every term in the 1st polynomial by every term in the 2nd polynomial Step 2: Combine Like Terms Step 3: Place in Decreasing Order of Exponent 4x2 (2x3 + 10x2 – 2x – 5) = 8x5 + 40x4 –8x3 –20x2 (x + 5) (2x3 + 10x2 – 2x – 5) = 2x4 + 10x3 – 2x2 – 5x + 10x3 + 50x2 – 10x – 25 = 2x4 + 20x3 + 48x2 –15x -25

13. 2x4 10x3 -2x2 -5x 10x3 50x2 -10x -25 Another Method for Multiplication Multiply:(x + 5) (2x3 + 10x2 – 2x – 5) 2x3 10x2 – 2x – 5 x 5 Answer: 2x4 + 20x3 +48x2 –15x -25

14. Binomial Multiplication with FOIL (2x + 3) (x - 7) F. O. I. L. (First) (Outside) (Inside) (Last) (2x)(x) (2x)(-7) (3)(x) (3)(-7) 2x2 -14x 3x -21 2x2 -14x + 3x -21 2x2 - 11x -21

15. 5.5 & 5.6: Review: Factoring Polynomials To factor a number such as 10, find out ‘what times what’ = 10 10 = 5(2) To factor a polynomial, follow a similar process. Factor: 3x4 – 9x3 +12x2 3x2 (x2 – 3x + 4) Another Example: Factor 2x(x + 1) + 3 (x + 1) (x + 1)(2x + 3)

16. Solving Polynomial Equations By Factoring Zero Product Property : If AB = 0 then A = 0 or B = 0 Solve the Equation: 2x2 + x = 0 Step 1: Factor x (2x + 1) = 0 Step 2: Zero Product x = 0 or 2x + 1 = 0 Step 3: Solve for X x = 0 or x = - ½ Question: Why are there 2 values for x???

17. Factoring Trinomials To factor a trinomial means to find 2 binomials whose product gives you the trinomial back again. Consider the expression: x2 – 7x + 10 (x – 5) (x – 2) The factored form is: Using FOIL, you can multiply the 2 binomials and see that the product gives you the original trinomial expression. How to find the factors of a trinomial: Step 1: Write down 2 parentheses pairs. Step 2: Do the FIRSTS Step3 : Do the SIGNS Step4: Generate factor pairs for LASTS Step5: Use trial and error and check with FOIL

18. Practice • Factor: • y2 + 7y –30 4. –15a2 –70a + 120 • 10x2 +3x –18 5. 3m4 + 6m3 –27m2 • 8k2 + 34k +35 6. x2 + 10x + 25

19. 5.7 Special Types of Factoring Square Minus a Square A2 – B2 = (A + B) (A – B) Cube minus Cube and Cube plus a Cube (A3 – B3) = (A – B) (A2 + AB + B2) (A3 + B3) = (A + B) (A2 - AB + B2) Perfect Squares A2 + 2AB + B2 = (A + B)2 A2 – 2AB + B2 = (A – B)2

20. 5.8 Solving Quadratic Equations • General Form of Quadratic Equation • ax2 + bx + c = 0 • a, b, c are real numbers & a 0 • A quadratic Equation: x2 – 7x + 10 = 0 a = _____ b = _____ c = ______ • Methods & Tools for Solving Quadratic Equations • Factor • Apply zero product principle (If AB = 0 then A = 0 or B = 0) • Quadratic Formula (We will do this one later) 1 -7 10 Example1:Example 2: x2 – 7x + 10 = 0 4x2 – 2x = 0 (x – 5) (x – 2) = 0 2x (2x –1) = 0 x – 5 = 0 or x – 2 = 0 2x=0 or 2x-1=0 + 5 + 5 + 2 + 2 2 2 +1 +1 2x=1 x = 5 or x = 2 x = 0 or x=1/2

21. Solving Higher Degree Equations x3 = 4x x3 - 4x = 0 x (x2 – 4) = 0 x (x – 2)(x + 2) = 0 x = 0 x – 2 = 0 x + 2 = 0 x = 2 x = -2 2x3 + 2x2 - 12x = 0 2x (x2 + x – 6) = 0 2x (x + 3) (x – 2) = 0 2x = 0 or x + 3 = 0 or x – 2 = 0 x = 0 or x = -3 or x = 2

22. Solving By Grouping x3 – 5x2 – x + 5 = 0 (x3 – 5x2) + (-x + 5) = 0 x2 (x – 5) – 1 (x – 5) = 0 (x – 5)(x2 – 1) = 0 (x – 5)(x – 1) (x + 1) = 0 x – 5 = 0 or x - 1 = 0 or x + 1 = 0 x = 5 or x = 1 or x = -1

23. B c a A C b Pythagorean Theorem Right Angle – An angle with a measure of 90° Right Triangle – A triangle that has a right angle in its interior. Hypotenuse Pythagorean Theorem a2 + b2 = c2 (Leg1)2 + (Leg2)2 = (Hypotenuse)2 Legs