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1 4.1 Exponents n is your power; x is your base Read x to the nth power 3 factors of x, 5 factors of y xxxyyyyy What is the coefficient for the above term? What is the power on this term? 3x

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2 Rules of Exponents: Product Rule: When multiplying like bases, add the exponents.

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3 Rules of Exponents: Quotient Rule: When dividing like bases, subtract the exponents.

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4 Rules of Exponents: Zero Exponent Rule: Any number raised to the zero power = 1

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5 Rules of Exponents: Power Rule: When taking a power to a power, multiply exponents.

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6 Rules of Exponents: Expanded Power Rule: When taking a single term to a power, you must take each part to the power.

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7 Rules of Exponents: If you forget one of the rules, make up an easy problem that you know the answer to. From this problem, you can reinvent the rule for yourself:

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8 Negative Exponent Rule Fraction raised to a negative exponent rule

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9 4.2 Negative Exponents Negative Exponent Rule: A negative exponent sends the base to the denominator. If it is already in the denominator, it will actually come back up to the top. When simplifying, your final answer should have NO negative exponents.

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10 4.3 Scientific Notation Scientific Notation is a way to write VERY large or VERY small numbers Scientific Notation always has TWO parts: 1) A number between 1 and 10 (number must be > 1 AND < 10) 2) A power of ten

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11 4.3 Scientific Notation Examples of numbers and their equivalent scientific notation form: 107,500,000 = 1.075 x 10 8 0.000756 = 7.56 x 10 -4

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12 4.3 Scientific Notation To put a number that is in standard notation INTO scientific notation, 1)Drop a decimal point in to create a number between 1 and 10 107,500,000 = drop a decimal between 1 and 0 = 1.075 0.000756 = drop a decimal in between 7 and 5 = 7.56

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13 4.3 Scientific Notation 2)Count the number of places you need to move the decimal, noting the direction as well. (positive numbers go to the right and negative numbers to the left) 1.075 x 10 ? How many places do I need to move to get back to 107,500,000? 1.075 x 10 8 7.56 x 10 ? How many places do I need to move to get back to 0.000756? 7.56 x 10 -4

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14 4.3 Multiplying or Dividing numbers in Scientific Notation You can not add or subtract numbers when they are in scientific notation, but you can multiply or divide them. And it is not hard either – just multiply or divide the corresponding parts. (3.6 x 10 -2 ) x (2.0 x 10 7 ) = 7.2 x 10 5 (3.6 x 10 -2 ) / (2.0 x 10 7 ) = 1.8 x 10 -9

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15 4.3 Scientific Notation Be careful! Sometimes your answer will not be in proper scientific notation: 57 x 10 6 This is not proper scientific notation because 57 is not between 1 and 10. Move the decimal over and adjust your power: 5.7 x 10 7 If you move your dec point to the left, you add one to your power of ten. If you move your dec point to the right, you subtract one from your power of ten.

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16 4.4 Addition/Subtraction of Polynomials A Polynomial is an expression containing the sum of a finite number of terms containing x 3x 2 + 5x 3 – 5 + 12x 7 Polynomials are normally written in descending order of the variable: highest exponent on the variable first and down from there; Constant is always last (5x 0 ) 12x 7 + 5x 3 + 3x 2 – 5

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17 4.4 Addition/Subtraction of Polynomials Polynomial-general term for these expressions Monomial-1 term6x Binomial-2 terms6x + 8 Trinomial-3 terms6x 2 – 8x + 4

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18 4.4 Addition/Subtraction of Polynomials Degree of a term-exponent of the variable in that term 6x 2 – 8x + 4x 2 y + x 2 y 3 - x 2 1 0 3 5 1 Degree of a polynomial-is the same as that of its highest-degree term 6x 2 – 8x + 4x 2 y + x 2 y 3 - x 2 nd degree 5 th degree

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19 4.4 Addition/Subtraction of Polynomials To add polynomials, combine like terms. Remember that like terms have the same variables and the same degrees of those variables. To subtract polynomials, use the distributive property to remove parentheses (change every sign in the parenthesis of the polynomials being subtracted) and then combine like terms. Show columns too.

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20 4.5 Multiplication of Polynomials Monomial x monomial (4x 2 )(5x 3 ) = 20x 5 Monomial x polynomial (distribute) 4x 2 (5x 3 +2x 2 +5x – 7)= 20x 5 +8x 4 +20x 3 – 28x 2 Binomial x binomial (FOIL) (x+5)(x-7)=x 2 - 7x + 5x – 35=x 2 – 2x - 35

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21 4.5 Multiplication of Polynomials Difference of Squares -watch the middle term drop out. (a+b)(a-b) = a 2 + ab – ab – b 2 = a 2 – b 2 (x+4)(x-4) = x 2 +4x – 4x – 16 = x 2 - 16 Square of binomial formulas (a+b) 2 = (a+b)(a+b)=a 2 +2ab+b 2 (a-b) 2 = (a-b)(a-b)=a 2 -2ab+b 2

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22 4.5 Multiplication of Polynomials Any two polynomials can be multiplied together by distributing each term of the 1 st through the 2nd (x 2 + 3x + 7)(4x 3 + x 2 – 7x – 2) 4x 5 + 1x 4 – 7x 3 – 2x 2 12x 4 + 3x 3 – 21x 2 – 6x 28x 3 + 7x 2 – 49x – 14 4x 5 + 13x 4 + 24x 3 – 16x 2 – 55x - 14

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23 4.6 Division of Polynomials Short division: Divide a polynomial by a monomial- -divide each term of the polynomial by the monomial 10x 2 – 4x10x 2 - 4x5x - 2 2x 2x 2x

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24 4.6 Division of Polynomials Long division-divide a polynomial by a binomial- (x 2 + 3x + 2) ÷ (x + 1) x + 2 x + 1 x 2 + 3x + 2 -(x 2 + 1x) 2x + 2 -(2x + 2) 0 No remainder—if you had a remainder, what would you do with it?

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25 4.6 Division of Polynomials Long division-divide a polynomial by a binomial- (x 2 + 3x + 7) ÷ (x + 1) x + 2 x + 1 x 2 + 3x + 7 -(x 2 + 1x) 2x + 7 -(2x + 2) 5 X + 2 +

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26 4.6 Division of Polynomials To check your answer, multiply the divisor and the quotient (plus the remainder if you have one) and it will equal the dividend Quotient + Remainder DivisorDividend

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27 4.6 Division of Polynomials If you are missing a term when the polynomials are written in descending order, place a zero where the missing term should go – as a place holder. (9x 2 - 16) ÷ (3x - 4) 3x + 4 3x - 4 9x 2 + 0x - 16 -(9x2 - 12x) 12x - 16 -(12x - 16) 0

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