5.1&5.2 Exponents 8 2 =8 8 = = = 16 x 2 = x xx 4 = x x x xBase = x Exponent = 2Exponent = 4 Exponents of 1Zero Exponents Anything to the 1 power is itself Anything to the zero power = = 5 x 1 = x (xy) 1 = xy5 0 = 1 x 0 = 1 (xy) 0 = 1 Negative Exponents 5 -2 = 1/(5 2 ) = 1/25 x -2 = 1/(x 2 ) xy -3 = x/(y 3 ) (xy) -3 = 1/(xy) 3 = 1/(x 3 y 3 ) a -n = 1/a n 1/a -n = a n a -n /a -m = a m /a n

Powers with Base = = = = = 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 10 0 = = 1/10 1 = 1/10 = = 1/10 2 = 1/100 = = 1/10 3 = 1/1000 = = 1/10 4 = 1/10000 =.0001 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

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) Changing a number from scientific notation to standard form Step 1: Write the number down without the 10 n 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 10 n where n is the # of digits moved left adding a x10 -n where n is the #digits moved right = x 10 -2

Raising Quotients to Powers a n b = anbnanbn a -n b = a- n b- n = bnanbnan = b n a Examples: == 2x 3 (2x) 3 8x 3 y y 3 y 3 = = 2x -3 (2x) -3 1 y 3 y 3 y y -3 y -3 (2x) 3 (2x) 3 8x 3 == ==

Product Rule a m a n = a (m+n) x 3 x 5 = xxx xxxxx = x 8 x -3 x 5 = xxxxx = x 2 = x 2 xxx 1 x 4 y 3 x -3 y 6 = xxxxyyyyyyyyy = xy 9 xxx 3x 2 y 4 x -5 7x = 3xxyyyy 7x = 21x -2 y 4 = 21y 4 xxxxx x 2

Quotient Rule a m = a (m-n) a n 4 3 = = 4 1 = = 64 = 8 = x 5 = xxxxx = x 3 x 5 = x (5-2) = x 3 x 2 xx x 2 15x 2 y 3 = 15 xx yyy = 3y 2 15x 2 y 3 = 3 x -2 y 2 = 3y 2 5x 4 y 5 xxxx y x 2 5x 4 y x 2 3a -2 b 5 = 3 bbbbb bbb = b 8 3a -2 b 5 = a (-2-4) b (5-(-3)) = a -6 b 8 = b 8 9a 4 b -3 9aaaa aa 3a 6 9a 4 b a 6

Powers to Powers (a m ) n = a mn (a 2 ) 3 a 2 a 2 a 2 = aa aa aa = a 6 (2 4 ) -2 = 1 = 1 = 1 = 1/256 ( 2 4) (x 3 ) -2 = x –6 = x 10 = x 4 (x -5 ) 2 x –10 x 6 (2 4 ) -2 = 2 -8 = 1 = 1

Products to Powers (ab) n = a n b n (6y) 2 = 6 2 y 2 = 36y 2 (2a 2 b -3 ) 2 = 2 2 a 4 b -6 = 4a 4 = a 4(ab 3 ) 3 4a 3 b 9 4a 3 b 9 b 6 b 15 What about this problem? 5.2 x = 5.2/3.8 x 10 9  1.37 x x 10 5 Do you know how to do exponents on the calculator?

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

5.3 Polynomials TERM a number: 5 a variable X a product of numbers and variables raised to powers 5x 2 y 3 p x (-1/2) y -2 z MONOMIAL -- Terms in which the variables have only nonnegative integer exponents. -4 5y x 2 5x 2 z 6 -xy 7 6xy 3 A coefficient is the numeric constant in a monomial. 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. POLYNOMIAL - A Monomial or a Sum of Monomials: 4x 2 + 5xy – y 2 (3 Terms) Binomial – A polynomial with 2 Terms (X + 5) Trinomial – A polynomial with 3 Terms

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

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

Another Method for Multiplication Multiply: (x + 5) (2x x 2 – 2x – 5) 2x 3 10x 2 – 2x – 5 x5x5 2x 4 10x 3 -2x 2 -5x 10x 3 50x 2 -10x -25 Answer: 2x x 3 +48x 2 –15x -25

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

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

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

Factoring Trinomials To factor a trinomial means to find 2 binomials whose product gives you the trinomial back again. Consider the expression: x 2 – 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

Practice Factor: 1.y 2 + 7y –304. –15a 2 –70a x 2 +3x –185. 3m 4 + 6m 3 –27m 2 3.8k k x x + 25

5.7 Special Types of Factoring Square Minus a Square A 2 – B 2 = (A + B) (A – B) Cube minus Cube and Cube plus a Cube (A 3 – B 3 ) = (A – B) (A 2 + AB + B 2 ) (A 3 + B 3 ) = (A + B) (A 2 - AB + B 2 ) Perfect Squares A 2 + 2AB + B 2 = (A + B) 2 A 2 – 2AB + B 2 = (A – B) 2

5.8 Solving Quadratic Equations General Form of Quadratic Equation ax 2 + bx + c = 0 a, b, c are real numbers & a  0 A quadratic Equation: x 2 – 7x + 10 = 0a = _____ b = _____ c = ______ Methods & Tools for Solving Quadratic Equations 1.Factor 2.Apply zero product principle (If AB = 0 then A = 0 or B = 0) 3.Quadratic Formula (We will do this one later) Example1: Example 2: x 2 – 7x + 10 = 04x 2 – 2x = 0 (x – 5) (x – 2) = 02x (2x –1) = 0 x – 5 = 0 or x – 2 = 02x=0 or 2x-1= x=1 x = 5 or x = 2 x = 0 or x=1/

Solving Higher Degree Equations x 3 = 4x x 3 - 4x = 0 x (x 2 – 4) = 0 x (x – 2)(x + 2) = 0 x = 0 x – 2 = 0 x + 2 = 0 x = 2 x = -2 2x 3 + 2x x = 0 2x (x 2 + 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

Solving By Grouping x 3 – 5x 2 – x + 5 = 0 (x 3 – 5x 2 ) + (-x + 5) = 0 x 2 (x – 5) – 1 (x – 5) = 0 (x – 5)(x 2 – 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

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