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Transcript
Solving Polynomial Equations To solve polynomials equations by factoring. To solve polynomials equations by graphing. Use graphing technology to find approximate solutions for polynomial equations. Use polynomials equations to solve real world problems. 3 3 2 2 a b a b a ab b Sum of the cubes= Difference of the cubes= a 3 b3 a ba 2 ab b 2 Remember: If x a is a factor of polynomial, then the polynomial has a value 0 when x=a. If a is a real number, then the graph of the polynomial has (a,0), as an x-intercept. 2 When you have a polynomial ax bx c factoring that polynomial will help you to solve the 2 equation like ax bx c 0 So to solve a polynomial equation by factoring Write the equation in the form P(x)=0 Factor P(x). Use the Zero Property of the roots Problem 1: Solving Polynomial Equation Using Factors. What are the real or imaginary solutions of each polynomial equations? A. 2 x 3 5 x 2 3 x Make equation=0 and get GCF B. 3 x 4 12 x 2 6 x 3 Make equation =0, Divide by 3 and get GCF 2 x 3 5 x 2 3x 0 3 x 4 6 x 3 12 x 2 0 x 2 x 2 5x 3 0 x 4 2 x3 4 x 2 0 2 x 2x 6x x 3 2 2 x( x 3 ) ( x 3 ) ( x 3 )( 2 x 1 ) Zero property and solve for x x 2 x 1 x 3 0 1 x 0, x , x 3 2 5x 3 2 x2 5 x 3 factor ( 2 3 6 ) Factor 2 x2 x2 2x 4 0 Use quadratic to solve x 2 x 2 2 x 4 0 x2 0 x 0 x 2 22 414 21 2 12 2 2i 3 2 2 1 i 3 x 0, x 1 i 3 , x 1 i 3 Your turn What are the real or imaginary solutions of each equation? A. x 2 1 x 2 4 0 x 2 1 applied B. x 5 4 x 3 5 x 4 2 x 3 Answers: x 1 x 4 2 applied x 2 i x 1, x 1, x 2i , x 2i x5 5x4 6x3 0 x3 x2 5x 6 0 x 3 x 3 x 2 0 x 0, x 2, x 3 Take a note Problem 2: Solving Polynomial Equations by Factoring What are the real or imaginary solutions of each polynomial equation? 4 2 x 3 x 4 A. Make the equation =0 x4 3x2 4 0 Let a x 2 then a 2 3a 4 0 Factor with -4 and +1 a 4a 1 0 x 2 4x 2 1 0 Replace back x2 x20 x -2 ax 2 2 2 2 Then x 4 x 2 x 2 x 2 x 2 x 2x 2 1 0 x20 x 4 a 2 and 3 x 2 3a x2 1 0 and x 2 1 applied x i Two imaginary roots Verify your answer with the graphic calculator The graphs shows zero at x=2, and x=-2. it also shows three turning points. This means that are imaginary roots, which do not appear on the graph Problem 2 What are the real or imaginary solutions of each polynomial equation? B. x 3 1 x 1 0 3 Make equation =0 Factor the difference of cubes x 1x 2 x 1 0 x 1 0 x 1 x x 1 0 1 12 411 1 i 3 x 21 2 2 use Q.F The three solutions are 1, 1 i 3 , 1 i 3 2 2 2 2 Your turn What are the real or imaginary solutions of each 3 x 8x 8x 8 polynomial equations? 3 x 8x 8x 8 0 a ) x 4 16 Answer: c) b) x 8 x 2 x 3 x 3 23 0 2 x 2x 2 2 x 2 2 0 2 2 2 414 x 21 c ) x x 2 8 8 x 1 Answer: a) x 2 22 x 2 22 0 x 2 x 2x x 2 x 2i 2 4 0 Answer b) x 1 i 3 x3 2x2 8x 0 x x2 2x 8 0 x x 4 x 2 0 x 0, x 4, x 2 Problem 3: Finding Real Roots by Graphing What are the real solutions of the equation x 3 5 4 x 2 x Method 1: graph y1 x 3 5 and y2 4 x 2 x Use the intersect feature to find the x values of the points of intersection. Approximate solutions are x=-1.09,x=1.16, and x=3.93 Method 2: Rewrite the equation as Graph the related function. x3 4x2 x 5 0 Use the zero feature 3 2 y x 4x x 5 The solutions are the same that in method 1, so the approximate solutions are x=-1.09,x=1.16, and x=3.93 Verify the solutions by showing that they satisfy the original equation. Show values of 3 2 y1 x 5 and y2 4 x x in a table. Your turn: a)What are the real solutions of the equation. b)Which method seems to be easier and more reliable way to find the solutions of an equations? Explain x3 x2 x 1 Answer a) -1.84 b) the second method seems to be a more reliable way to find the solutions because you do not risk missing a point of intersection Problem 4:Modeling a real world situation Close friends Stacy, Una, and Amir were all born on July 4. Stacy is one year younger than Una. Una is two years younger than Amir. On July 4, 2010, the product of their ages was 2300 more than the sum of their ages. How old was each friend on that day? Let x=Una’s age on July 4,2010 Stacy’s age=x-1 Amir’s age=x+2 Answer Sum of their ages x+(x-1)+(x+2)+2300 3 x 2301 x( x x 2) 2 3 x 2301 x x 2 x 3 2 x x 5 x 2301 0 3 2 = Product of their ages x(x-1)(x+2) Graph it and look into the tabl when y=0 and take only the solution that makes sense x=13. Una was 13,Stacy 12 and Amir 15 Your turn A. What are three consecutive integers whose product is 480 more than their sum. Answer: 7,8,9 x x 1 x 2 480 x x 1 x 2 x 2 x x 2 480 3x 3 x 2 x x 2 x 483 3x 3 2 2 x3 3x 2 2 x 483 3x x3 3x 2 x 483 0 Input the data in the function feature and look for the table when y=0. B. What are three consecutive even integers whose product is 4 times their sum? Answer: 2, 4, 6 or -6, -4, -2 x x 2 x 4 4 x x 2 x 4 x 3 6 x 2 4 x 24 0 Classwork odd Homework even TB pgs 301 exercises 1-56