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Transcript
Entry Task I am greater than my square. The sum of my numerator and denominator is 5 My numerator is a factor of 6 My denominator is a factor of 4 What fraction am I? How did you find me? 5.5 Theorems About Roots of Polynomial Equations Target: I can solve equations using the Rational Root Theorem I can use the Conjugate Root Theorem Let’s explore the following . . . (x + 2)(x - 3)(x - 4) = 0 The standard form of the equation is: x3 – 5x2 – 2x + 24 = 0 The roots are: -2, 3, 4 Take a closer look at the original equation and our roots: x3 – 5x2 – 2x + 24 = 0 The roots are: -2, 3, 4 What do you notice? -2, 3, and 4 all go into the last term, 24! Let’s look at another The standard form of the equation is: The roots are: Take a closer look at the original equation and our roots: 24x3 – 22x2 – 5x + 6 = 0 This equation factors to: (x + 1)(x - 2)(x - 3)= 0 2 3 4 The roots therefore are: -1, 2, 3 2 3 4 What do you notice? The numerators 1, 2, and 3 all go into the last term, 6! The denominators (2, 3, and 4) all go into the first term, 24! This leads us to the Rational Root Theorem 1. For polynomial x 3 x 2 3x 3 0 Here p = -3 and q = 1 Factors of -3 ±3, ±1 Or 3,-3, 1, -1 Factors of 1 ±1 Possible roots are ___________________________________ 3 2 2. For polynomial 3 x 9 x 4 x 12 0 Here p = 12 and q = 3 Factors of 12 ±12, ±6 , ±3 , ± 2 , ±1 ±4 Factors of 3 ±1 , ±3 Possible roots are ______________________________________________ Or ±12, ±4, ±6, ±2, ±3, ±1, ± 2/3, ±1/3, ±4/3 Wait a second . . . Where did all of these come from??? Let’s look at our solutions ±12, ±6 , ±3 , ± 2 , ±1, ±4 ±1 , ±3 Note that + 2 is listed twice; we only consider it as one answer Note that + 1 is listed twice; we only consider it as one answer Note that + 4 is listed twice; we only consider it as one answer 12 12 1 6 6 1 3 3 1 2 2 1 1 1 1 4 4 1 12 4 3 6 2 3 3 1 3 2 2 3 3 1 1 3 3 4 4 3 3 That is where our 9 possible answers come from! Let’s Try One Find the POSSIBLE roots of 5x3 - 24x2 + 41x - 20 = 0 Let’s Try One 5x3-24x2+41x-20=0 That’s a lot of answers! Obviously 5x3 - 24x2 + 41x – 20 = 0 does not have all of those roots as answers. Remember: these are only POSSIBLE roots. We take these roots and figure out what answers actually WORK. Step 1 – find p and q p = -3 q=1 Step 2 – by RRT, the only rational root is of the form… Factors of p Factors of q Step 3 – factors Factors of -3 = ±3, ±1 Factors of 1 = ± 1 Step 4 – possible roots -3, 3, 1, and -1 Step 5 – Test each root X -3 3 Step 6 – synthetic division X³ + X² – 3x – 3 (-3)³ + (-3)² – 3(-3) – 3 = -12 -1 1 1 -3 -3 -1 0 3 0 -3 0 (3)³ + (3)² – 3(3) – 3 = 24 1 (1)³ + (1)² – 3(1) – 3 = -4 -1 (-1)³ + (-1)² – 3(-1) – 3 = 0 1 THIS IS YOUR ROOT BECAUSE WE ARE LOOKING FOR WHAT ROOTS WILL MAKE THE EQUATION =0 1x² + 0x -3 Step 7 – Rewrite x³ + x² - 3x - 3 = (x + 1)(x² – 3) Step 8– factor more and solve (x + 1)(x² – 3) (x + 1)(x – √3)(x + √3) Roots are -1, ± √3 • Step 1 – find p and q • p = -6 • q=1 • Step 2 – by RRT, the only rational root is of the form… • Factors of p Factors of q • Step 3 – factors • Step 4 – possible roots • Factors of -6 = ±1, ±2, ±3, ±6 Factors of 1 = ±1 • -6, 6, -3, 3, -2, 2, 1, and -1 • Step 5 – Test each root X • Step 6 – synthetic division x³ – 5x² + 8x – 6 -6 -450 6 78 3 0 THIS IS YOUR ROOT -3 -102 2 -2 -2 -50 1 -2 -1 -20 3 1 1 -5 8 -6 3 -6 6 -2 2 0 1x² + -2x + 2 • Step 7 – Rewrite • x³ – 5x² + 8x – 6 = (x - 3)(x² – 2x + 2) • Step 8– factor more and solve • (x - 3)(x² – 2x + 2) X= 3 Quadratic Formula x 1 i • Roots are 3, 1 ± i Irrational Root Theorem • • • • Y a0 x n a1x n1 ... an1x an For a polynomial If a + √b is a root, Then a - √b is also a root Irrationals always come in pairs. Real values do not. Complex pairs of form a + √ b and a - √ b CONJUGATE ___________________________ 1. For polynomial has roots 3 + √2 2 3 - √2 Degree of Polynomial ______ Other roots ______ 2. For polynomial has roots -1, 0, - √3, 1 + √5 6 √3 , 1 - √5 Degree of Polynomial ______ Other roots __________ 1. For polynomial has roots 1 + √3 and -√11 1 - √3 Other roots ______ √11 _______ 4 Degree of Polynomial ______ Question: One of the roots of a polynomial is Can you be certain that 4 2 4 2 is also a root? No. The Irrational Root Theorem does not apply unless you know that all the coefficients of a polynomial are rational. You would have to have as your root to make use of the IRT. 4 2 Write a polynomial given the roots 5 and √2 • Another root is - √2 • Put in factored form • y = (x – 5)(x + √2 )(x – √2 ) Homework • P. 316 12,14,18,19,21,25,28,33,34,38 Descartes' Rule of Signs 5 Decide what to FOIL first y = (x – 5)(x + √2 )(x – √2 ) X x √2 X2 X √2 -√2 -X √2 -2 (x² – 2) FOIL or BOX to finish it up (x-5)(x² – 2) y = x³ – 2x – 5x² + 10 Standard Form y = x³ – 5x² – 2x + 10 x2 x -5 -2 X3 -2x -5x2 10 Write a polynomial given the roots -√5, √7 • • • • Other roots are √5 and -√7 Put in factored form y = (x – √5 )(x + √5)(x – √7)(x + √7) Decide what to FOIL first y = (x – √5 )(x + √5)(x – √7)(x + √7) Foil or use a box method to multiply the binomials X x √5 X2 X √5 -√5 -X √5 -5 (x² – 5) X x √7 X2 X √7 -√7 -X √7 -7 (x² – 7) y = (x² – 5)(x² – 7) x2 x2 FOIL or BOX to finish it up y = x4 – 7x² – 5x² + 35 Clean up y = x4 – 12x² + 35 -7 -5 X4 -5x2 -7x2 35