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A rational number is A rational number is a number that can be turned into a fraction A rational number is 2 is rational a number that can be turned into a fraction A rational number is 2 is rational because it could be turned into a number that can be turned into a fraction 2 1 A rational number is 0.5 is rational 2 is rational because it could be turned into a number that can be turned into a fraction 2 1 A rational number is 0.5 is rational 2 is rational because it could be turned into a number that can be turned into a fraction because it could be turned into 1 2 2 1 In general, rational numbers are anything BUT In general, rational numbers are anything BUT square roots In general, rational numbers are anything BUT So √2 is not rational square roots In general, rational numbers are anything BUT So √2 is not rational but √4 is square roots In general, rational numbers are anything BUT So √2 is not rational but √4 is square roots because it equal 2 √5 .23 Irrational ? Rational ? A 'root' is an answer to a polynomial equation A 'root' is an answer to a polynomial equation x2 - 2x - 3 = 0 A 'root' is an answer to a polynomial equation x2 - 2x - 3 = 0 This is a polynomial A 'root' is an answer to a polynomial equation x2 - 2x - 3 = 0 This is a polynomial the equal sign makes this an equation A 'root' is an answer to a polynomial equation x2 - 2x - 3 = 0 So this is a polynomial equation The answers to this equation are 3 and -1 A 'root' is an answer to a polynomial equation x2 - 2x - 3 = 0 The answers to this equation are 3 and -1 A 'root' is an answer to a polynomial equation these are also called x2 - 2x - 3 = 0 The answers to this equation are 3 and -1 A 'root' is an answer to a polynomial equation these are also called x2 - 2x - 3 = 0 the roots The answers to this equation are 3 and -1 The reason it is called 'roots' is complicated A 'root' is an answer to a polynomial equation these are also called x2 - 2x - 3 = 0 the roots The answers to this equation are 3 and -1 The reason it is called 'roots' is complicated A 'root' is an answer to a polynomial equation but it has to do with the idea that the solution these are also called x2 - 2x - 3 = 0 the roots The answers to this equation are 3 and -1 The reason it is called 'roots' is complicated A 'root' is an answer to a polynomial equation but it has to do with the idea that the solution these are also called to the equation x2=9 x2 - 2x - 3 = 0 the roots The answers to this equation are 3 and -1 The reason it is called 'roots' is complicated A 'root' is an answer to a polynomial equation but it has to do with the idea that the solution these are also called to the equation x2=9 would be the square ROOTS of 9 (3 and -3) x2 - 2x - 3 = 0 the roots The answers to this equation are 3 and -1 The reason it is called 'roots' is complicated A 'root' is an answer to a polynomial equation but it has to do with the idea that the solution these are also called to the equation x2=9 would be the square ROOTS of 9 (3 and -3) x2 - 2x - 3 = 0 the roots and so the answers could be called roots as well The roots/answers/solutions also happen to be The roots/answers/solutions also happen to be the x-intercepts We already meantioned that the roots of The roots/answers/solutions also happen to be the x-intercepts x2 - 2x - 3 = 0 We already meantioned that the roots of The roots/answers/solutions also happen to be the x-intercepts x2 - 2x - 3 = 0 are 3 and -1 We already meantioned that the roots of The roots/answers/solutions also happen to be the x-intercepts x2 - 2x - 3 = 0 are 3 and -1 the x intercepts We already meantioned that the roots of The roots/answers/solutions also happen to be the x-intercepts x2 - 2x - 3 = 0 are 3 and -1 the x intercepts We already meantioned that the roots of The roots/answers/solutions also happen to be the x-intercepts x2 - 2x - 3 = 0 are 3 and -1 the x intercepts We already meantioned that the roots of The roots/answers/solutions also happen to be are also 3 and -1 the x-intercepts x2 - 2x - 3 = 0 are 3 and -1 Answer listed smallest to largest with comma and space betweeneach answer (e.g. -4, 12) The roots are The rational root theorum gives us a list The rational root theorum gives us a list of all the possible rational roots The rational root theorum gives us a list of all the possible rational roots of a polynomial Given the polynomial 5x3-7x2-5x+7 The rational root theorum gives us a list of all the possible rational roots of a polynomial Given the polynomial 5x3-7x2-5x+7 The rational root theorum gives us a list write a fraction bar of all the possible rational roots of a polynomial Given the polynomial 5x3-7x2-5x+7 The rational root theorum gives us a list write a fraction bar of all the possible rational roots of a polynomial write all the factors of the last number Given the polynomial 5x3-7x2-5x+7 The rational root theorum gives us a list write a fraction bar of all the possible rational roots of a polynomial write all the factors of the last number Given the polynomial 5x3-7x2-5x+7 The rational root theorum gives us a list write a fraction bar of all the possible rational roots of a polynomial write all the factors of the last number Given the polynomial 5x3-7x2-5x+7 The rational root theorum gives us a list write a fraction bar of all the possible rational roots of a polynomial 1 write all the factors of the last number Given the polynomial 5x3-7x2-5x+7 The rational root theorum gives us a list write a fraction bar of all the possible rational roots of a polynomial 1 7 write all the factors of the last number Given the polynomial 5x3-7x2-5x+7 The rational root theorum gives us a list write a fraction bar then write all the factors of the first number of all the possible rational roots of a polynomial 1 7 write all the factors of the last number Given the polynomial 5x3-7x2-5x+7 The rational root theorum gives us a list write a fraction bar then write all the factors of the first number of all the possible rational roots of a polynomial 1 7 write all the factors of the last number Given the polynomial 5x3-7x2-5x+7 The rational root theorum gives us a list write a fraction bar then write all the factors of the first number of all the possible rational roots of a polynomial 1 1 7 write all the factors of the last number Given the polynomial 5x3-7x2-5x+7 The rational root theorum gives us a list write a fraction bar then write all the factors of the first number of all the possible rational roots of a polynomial 1 1 7 5 write all the factors of the last number Given the polynomial 5x3-7x2-5x+7 The rational root theorum gives us a list write a fraction bar then write all the factors of the first number of all the possible rational roots then write all the possible combinations of a polynomial 1 1 7 5 write all the factors of the last number Given the polynomial 5x3-7x2-5x+7 The rational root theorum gives us a list write a fraction bar then write all the factors of the first number of all the possible rational roots then write all the possible combinations of a polynomial 1, 1 1 7 5 write all the factors of the last number Given the polynomial 5x3-7x2-5x+7 The rational root theorum gives us a list write a fraction bar then write all the factors of the first number of all the possible rational roots then write all the possible combinations of a polynomial 1, 1/5, 1 1 7 5 write all the factors of the last number Given the polynomial 5x3-7x2-5x+7 The rational root theorum gives us a list write a fraction bar then write all the factors of the first number of all the possible rational roots then write all the possible combinations of a polynomial 1, 1/5, 7, 1 1 7 5 write all the factors of the last number Given the polynomial 5x3-7x2-5x+7 The rational root theorum gives us a list write a fraction bar then write all the factors of the first number of all the possible rational roots then write all the possible combinations of a polynomial 1, 1/5, 7, 7/5 1 1 7 5 Given the polynomial 5x3-7x2-5x+7 The rational root theorum gives us a list So the list of possible rational roots is of all the possible rational roots of a polynomial 1, 1/5, 7, 7/5 1 1 7 5 Given the polynomial 5x3-7x2-5x+7 The rational root theorum gives us a list So the list of possible rational roots is of all the possible rational roots +1 of a polynomial 1, 1/5, 7, 7/5 1 1 7 5 Given the polynomial 5x3-7x2-5x+7 The rational root theorum gives us a list So the list of possible rational roots is of all the possible rational roots +1, +1/5 of a polynomial 1, 1/5, 7, 7/5 1 1 7 5 Given the polynomial 5x3-7x2-5x+7 The rational root theorum gives us a list So the list of possible rational roots is of all the possible rational roots +1, +1/5, +7 of a polynomial 1, 1/5, 7, 7/5 1 1 7 5 Given the polynomial 5x3-7x2-5x+7 The rational root theorum gives us a list So the list of possible rational roots is of all the possible rational roots +1, +1/5, +7, +7/5 of a polynomial 1, 1/5, 7, 7/5 1 1 7 5 Given the polynomial 5x3-7x2-5x+7 The rational root theorum gives us a list So the list of possible rational roots is of all the possible rational roots +1, +1/5, +7, +7/5, -1 of a polynomial 1, 1/5, 7, 7/5 1 1 7 5 Given the polynomial 5x3-7x2-5x+7 The rational root theorum gives us a list So the list of possible rational roots is of all the possible rational roots +1, +1/5, +7, +7/5, -1, -1/5 of a polynomial 1, 1/5, 7, 7/5 1 1 7 5 Given the polynomial 5x3-7x2-5x+7 The rational root theorum gives us a list So the list of possible rational roots is of all the possible rational roots +1, +1/5, +7, +7/5, -1, -1/5, -7 of a polynomial 1, 1/5, 7, 7/5 1 1 7 5 Given the polynomial 5x3-7x2-5x+7 The rational root theorum gives us a list So the list of possible rational roots is of all the possible rational roots +1, +1/5, +7, +7/5, -1, -1/5, -7, -7/5 of a polynomial 1, 1/5, 7, 7/5 1 1 7 5 Factors of 8 1, , 4 , 3x3 -5x2 +x -4 Factors of 4 ? Factors of 3 ? 3x3 -5x2 +x -4 1, 2, 4 ? 1, 3 ? Factors of 4 Factors of 3 How do we know which numbers on the list How do we know which numbers on the list are actually answers/roots/solutions? There are three ways. How do we know which numbers on the list are actually answers/roots/solutions? There are three ways. How do we know which numbers on the list Plug In are actually answers/roots/solutions? There are three ways. How do we know which numbers on the list Plug In Divide are actually answers/roots/solutions? There are three ways. How do we know which numbers on the list Plug In Divide Graph are actually answers/roots/solutions? Using the polynomial 3x3 +13x2 +13x +3 Using the polynomial 3x3 +13x2 +13x +3 let's find the list of rational roots Using the polynomial 3x3 +13x2 +13x +3 let's find the list of rational roots and see which ones actally work Using the polynomial 3x3 +13x2 +13x +3 1 3 1 3 let's find the list of rational roots and see which ones actally work Using the polynomial 3x3 +13x2 +13x +3 1 3 1 3 let's find the list of rational roots and see which ones actally work = +1, Using the polynomial 3x3 +13x2 +13x +3 1 3 1 3 let's find the list of rational roots and see which ones actally work = +1, +1/3 Using the polynomial 3x3 +13x2 +13x +3 1 3 1 3 Already got a '1' let's find the list of rational roots and see which ones actally work = +1, +1/3 Using the polynomial 3x3 +13x2 +13x +3 1 3 1 3 let's find the list of rational roots and see which ones actally work = +1, +1/3, +3 Using the polynomial 3x3 +13x2 +13x +3 1 3 1 3 let's find the list of rational roots and see which ones actally work = +1, +1/3, +3, -1, -1/3, -3 Using the polynomial 3x3 +13x2 +13x +3 1 3 1 3 let's find the list of rational roots and see which ones actally work Let's see if -1 works by plugging it in for x = +1, +1/3, +3, -1, -1/3, -3 Using the polynomial 3x3 +13x2 +13x +3 1 3 1 3 let's find the list of rational roots and see which ones actally work Let's see if -1 works by plugging it in for x = 3x3+13x2 +13x +3 +1, +1/3, +3, -1, -1/3, -3 Using the polynomial 3x3 +13x2 +13x +3 1 3 1 3 let's find the list of rational roots and see which ones actally work Let's see if -1 works by plugging it in for x = 3x3+13x2 +13x +3 3(-1)3+13(-1)2+13(-1)+3 +1, +1/3, +3, -1, -1/3, -3 Using the polynomial 3x3 +13x2 +13x +3 1 3 1 3 let's find the list of rational roots and see which ones actally work Let's see if -1 works by plugging it in for x = 3x3+13x2 +13x +3 3(-1)3+13(-1)2+13(-1)+3 +1, +1/3, +3, -1, -1/3, -3 -3+13-13+3 = 0 Using the polynomial 3x3 +13x2 +13x +3 1 3 1 3 let's find the list of rational roots and see which ones actally work Let's see if -1 works by plugging it in for x = 3x3+13x2 +13x +3 3(-1)3+13(-1)2+13(-1)+3 +1, +1/3, +3, -1, -1/3, -3 -3+13-13+3 = 0 it works It equals zero so Using the polynomial 3x3 +13x2 +13x +3 1 3 1 3 let's find the list of rational roots and see which ones actally work Now let's see if -1 works by dividing = +1, +1/3, +3, -1, -1/3, -3 Using the polynomial 3x3 +13x2 +13x +3 1 3 1 3 -1 let's find the list of rational roots and see which ones actally work Now let's see if -1 works by dividing 3 = 13 +1, +1/3, +3, -1, -1/3, -3 13 3 Using the polynomial 3x3 +13x2 +13x +3 1 3 1 3 -1 let's find the list of rational roots and see which ones actally work Now let's see if -1 works by dividing 3 3 = 13 +1, +1/3, +3, -1, -1/3, -3 13 3 Using the polynomial 3x3 +13x2 +13x +3 1 3 1 3 -1 let's find the list of rational roots and see which ones actally work Now let's see if -1 works by dividing 3 3 = 13 -3 +1, +1/3, +3, -1, -1/3, -3 13 3 Using the polynomial 3x3 +13x2 +13x +3 1 3 1 3 -1 let's find the list of rational roots and see which ones actally work Now let's see if -1 works by dividing 3 3 = 13 10 -3 +1, +1/3, +3, -1, -1/3, -3 13 3 Using the polynomial 3x3 +13x2 +13x +3 1 3 1 3 -1 let's find the list of rational roots and see which ones actally work Now let's see if -1 works by dividing 3 3 = 13 10 -3 +1, +1/3, +3, -1, -1/3, -3 13 -10 3 3 Using the polynomial 3x3 +13x2 +13x +3 1 3 1 3 -1 let's find the list of rational roots and see which ones actally work Now let's see if -1 works by dividing 3 3 = 13 10 -3 +1, +1/3, +3, -1, -1/3, -3 13 -10 3 -3 3 0 Using the polynomial 3x3 +13x2 +13x +3 1 3 1 3 -1 let's find the list of rational roots and see which ones actally work Now let's see if -1 works by dividing 3 3 = 13 10 -3 +1, +1/3, +3, -1, -1/3, -3 13 -10 3 -3 3 0 The remainder so it works is zero Now let's see if it works by graphing Now let's see if it works by graphing 3x3+13x2 +13x +3 1 -1 2 -2 1 -1 2 -2 3 -3 Now let's see if it works by graphing 3x3+13x2 +13x +3 1 -1 2 -2 1 -1 2 -2 3 -3 Now let's see if it works by graphing 3x3+13x2 +13x +3 -1 so it works An x-intercept is |