Mrs. Shahmoradian (Ms. Tanskley) Monday 2nd.  Answer the following questions about the Polynomial equation: 6x^3 + 8x^2 – 7x – 3 = 0 1) How many roots.

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Presentation transcript:

Mrs. Shahmoradian (Ms. Tanskley) Monday 2nd

 Answer the following questions about the Polynomial equation: 6x^3 + 8x^2 – 7x – 3 = 0 1) How many roots will this polynomial have? 2) Which of the following are roots of the polynomial? How do you know? (Hint: look back at your notes): 1, -3, ½, -1/3

Overall, good! A few things to remember: 1) When you are finding P(c) by synthetic substitution, the answer is the remainder. 2) In order to prove if x – 1 is a factor, use either synthetic substitution or ______ and look for zero! (A remainder of zero or P(c) = 0 means that x – c is a factor). 3) Degree = # of roots (Which theorem tells us this?) Remember to divide by the factor you know first before you find the remaining factors. Using the quadratic formula will not work on a degree 3 polynomial. It is called the quadratic formula because it works on quadraic (Degree 2) polynomials. 4) IF you know roots, you know factors. If you know factors, you know the equation. What is a factor? How does it relate to an equation? If you still have questions, be sure to get them answered before the test on Friday!! (Tutorial today and Thursday).

 Find the factors of the polynomial 6x^3 + 8x^2 – 7x – 3 = 0.  What information is missing? Can we find the factors?  Rational Root Theorem:  If a polynomial equation with integral coefficients has the root h/k, where h and k are relatively prime integers, then h must be a factor of the constant term of the polynomial and k must be a factor of the coefficient of the highest degree term.

 Objectives: Finding rational roots of polynomials.  Vocabulary: Rational- Can be written as a fraction.  Examples:  Done in class.  Example 2:  Solve the equation 2x^4 + 3x^3 – 7x^2 + 3x – 9 = 0. (Please finish writing down solution from pg. 382 in your textbook, labeled as example 1)  Tools and Rules: Rational Root Theorem: If a polynomial equation with integral coefficients has the root h/k, where h and k are relatively prime integers, then h must be a factor of the constant term of the polynomial and k must be a factor of the coefficient of the highest degree term. Example: 6x^3 + 8x^2 – 7x – 3 = 0 – 3 is the constant, so h must be a factor of -3 6x^3 is the highest degree term, so k must be a factor of 6

 Rational Root Theorem:  If a polynomial equation with integral coefficients has the root h/k, where h and k are relatively prime integers, then h must be a factor of the constant term of the polynomial and k must be a factor of the coefficient of the highest degree term.  Example:  6x^3 + 8x^2 – 7x – 3 = 0  – 3 is the constant, so h must be a factor of -3 : ±1, ±3  6x^3 is the highest degree term, so k must be a factor of 6: ±1, ±2, ±3, ±6  h/k = ±1, ±3/±1, ±2, ±3, ±6

 Solve the equation 2x^4 + 3x^3 – 7x^2 + 3x – 9 = 0.  Hint: Use the rational roots theorem, then check each combination to see which are the factors.  Question: Do you need to check all the factors?  Once we find one or two factors, we can use the quadratic equation to find the remaining ones (if we narrow the polynomial down to a 2 nd degree polynomial).

 See board (chalkboard examples pg. 382)

 P. 384: ORAL #2-10 evens (If you find rational roots, then solve the polynomial to find the missing roots). Pg. 385 Mixed Review #7-10 all