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Mrs. Shahmoradian (Ms. Tanskley) Monday 2nd
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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
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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).
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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.
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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
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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
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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).
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See board (chalkboard examples 1 + 2 pg. 382)
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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
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