HW4 Sol - Math 2312 Homework 4 Due: Wednesday, June 23,...

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Page 1 of 10 Math 2312 Homework 4 Due: Wednesday, June 23, 2010, at the beginning of class 1. Given that one of the roots of f ( x ) is -1, sketch the graph of the function using the leading coefficient and the multiplicity of the zeros: 3 5 6 8 6 2 ) ( x x x x f Solution: We note that 3 2 x is a factor: ) 4 3 ( 2 ) ( 2 3 3 x x x x f Since -1 is a root, then (x-(-1)) = (x+1) must be a factor of the polynomial 4 3 2 3 x x Applying synthetic division, 0 4 4 1 4 4 1 4 0 3 1 1 (with a place holder of 0 for the missing x term) we know that 1 -4 4 means that 4 4 1 2 x x is also a factor of the polynomial so that 2 3 3 2 3 ) 2 )( 1 ( 2 ) 2 )( 2 )( 1 ( 2 ) 4 4 )( 1 ( 2 ) ( x x x x x x x x x x x x f Note that the remaining quadratic 4 4 2 x x easily factors. The degree of f is 3+1+2=6 and the leading coefficient is negative, so that both “ends” of the graph must point upward. Also, the three roots are 0, -1, and 2. The zero root is multiplicity 3, so the graph must pass through the x-axis at this point. The -1 root is multiplicity 1, to the graph must also pass through the axis here. Finally, 2 is multiplicity 2, the graph must “bounce” here:
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Page 2 of 10 2. Divide: ) 2 3 ( ) 2 6 ( 2 2 4 x x x x Solution:     1 2 2 0 3 2 3 4 0 6 2 0 6 2 0 3 2 2 2 2 3 4 2 3 4 2 x x x x x x x x x x x x x x x Note: since we encountered a constant (1 in this case) in the quotient, we had to bring down the next TWO terms, the –x and the -2. So, 2 3 1 2 ) 2 3 ( ) 2 6 ( 2 2 2 2 4 x x x x x x x
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Page 3 of 10 3. Find all the zeros of the function (and multiplicity if greater than one). (a) 9 24 12 ) ( 2 t t t r Solution: The greatest common factor is 3,so ) 3 8 4 ( 3 ) ( 2 t t t r We need two factors of 4: either 4 (leading coefficient) : 4 and 1 or 2 and 2, along with the factors of 3 (constant term): 3 and 1 or -3 and -1. If we chose 2 and 2 for the factors of 4, then -1 and -3 would work for the factors of 3 since 2(-3)+(-1)(2)= -8 the x coefficient: ) 3 2 )( 1 2 ( 3 ) 3 8 4 ( 3 ) ( 2 t t t t t r The zeros occur when 2t-1=0 or t = ½
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HW4 Sol - Math 2312 Homework 4 Due: Wednesday, June 23,...

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