Explanation Given: The points are ( 0 , 42 ) , ( 1 , 0 ) , ( 2 , − 40 ) and ( 3 , − 72 ) . Calculation: The third degree polynomial function be, p ( x ) = a 0 + a 1 x + a 2 x 2 + a 3 x 3 Substituting x = 0 , 1 , 2 and 3 into p ( x ) and equating the results to the respective y -values produces the system of linear equations in the variables a 0 , a 1 , a 2 and a 3 as shown below. p ( 0 ) = a 0 + a 1 ( 0 ) + a 2 ( 0 ) 2 + a 3 ( 0 ) 3 = a 0 = 42 p ( 1 ) = a 0 + a 1 ( 1 ) + a 2 ( 1 ) 2 + a 3 ( 1 ) 3 = a 0 + a 1 + a 2 + a 3 = 0 p ( 2 ) = a 0 + a 1 ( 2 ) + a 2 ( 2 ) 2 + a 3 ( 2 ) 3 = a 0 + 2 a 1 + 4 a 2 + 8 a 3 = − 40 p ( 3 ) = a 0 + a 1 ( 3 ) + a 2 ( 3 ) 2 + a 3 ( 3 ) 3 = a 0 + 3 a 1 + 9 a 2 + 27 a 3 = − 72 The augmented matrix is, [ 1 0 0 0 42 1 1 1 1 0 1 2 4 8 − 40 1 3 9 27 − 72 ] Solving the system by using Gauss elimination with back substitution: − R 1 + R 2 → R 2 , − R 1 + R 3 → R 3 , − R 1 + R 4 → R 4 [ 1 0 0 0 42 0 1 1 1 − 42 0 2 4 8 − 82 0 3 9 27 − 114 ] R 3 − 2 R 2 To determine b) To graph: The polynomial function showing the points ( 0 , 42 ) , ( 1 , 0 ) , ( 2 , − 40 ) and ( 3 , − 72 ) .

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN: 9781305658004

Author: Ron Larson

Publisher: Cengage Learning

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Chapter 1.3, Problem 6E

To determine

a)

To calculate:

The polynomial function whose graph passes through the points (0,42),(1,0),(2,−40) and (3,−72).

To determine

b)

To graph:

The polynomial function showing the points (0,42),(1,0),(2,−40) and (3,−72).

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Chapter 1.3, Problem 5E

Chapter 1.3, Problem 7E

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a) To calculate: The polynomial function whose graph passes through the points ( 0 , 42 ) , ( 1 , 0 ) , ( 2 , − 40 ) and ( 3 , − 72 ) .

Solution Summary: The author calculates the third degree polynomial function whose graph passes through the points p(x)=42-41x-2x

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a)

To calculate:

The polynomial function whose graph passes through the points (0,42),(1,0),(2,−40) and (3,−72).

b)

To graph:

The polynomial function showing the points (0,42),(1,0),(2,−40) and (3,−72).

Want to see the full answer?

Chapter 1 Solutions

a) To calculate: The polynomial function whose graph passes through the points ( 0 , 42 ) , ( 1 , 0 ) , ( 2 , − 40 ) and ( 3 , − 72 ) .

Solution Summary: The author calculates the third degree polynomial function whose graph passes through the points p(x)=42-41x-2x

Videos

a) To calculate: The polynomial function whose graph passes through the points (0,42),(1,0),(2,−40) and (3,−72).

b) To graph: The polynomial function showing the points (0,42),(1,0),(2,−40) and (3,−72).

Want to see the full answer?

Chapter 1 Solutions

a)

To calculate:

The polynomial function whose graph passes through the points (0,42),(1,0),(2,−40) and (3,−72).

b)

To graph:

The polynomial function showing the points (0,42),(1,0),(2,−40) and (3,−72).