Explanation Given Information: The expression, ( a + b ) 4 Formula used: Binomial theorem is ( a + b ) = n n ! n ! ⋅ 0 ! a n + n ! ( n − 1 ) ! ⋅ 1 ! a ( n − 1 ) b + n ! ( n − 2 ) ! ⋅ 0 ! a ( n − 2 ) b 2 + ⋯ + n ! 0 ! ⋅ n ! b n Calculation: Expansion of the expression ( a + b ) 4 is calculated as follow, By using binomial theorem. ( a + b ) = 4 4 ! 4 ! ⋅ 0 ! a 4 + 4 ! ( 4 − 1 ) ! ⋅ 1 ! a ( 4 − 1 ) b + 4 ! ( 4 − 2 ) ! ⋅ 2 ! a ( 4 − 2 ) b 2 + 4 ! ( 4 − 3 ) ! ⋅ 3 ! a ( 4 − 3 ) b 3 4 ! 0 ! ⋅ 4 ! b 4 = 1 ⋅ a 4 + 4 ! ( 3 ) ! ⋅ 1 ! a ( 3 ) b + 4 ! ( 2 ) ! ⋅ 2 ! a ( 2 ) b 2 + 4 ! ( 1 ) ! ⋅ 3 ! a ( 1 ) b 3 + 4 ! 0 ! ⋅ 4 ! b 4 = a 4 + 4 × 3 × 2 × 1 3 × 2 × 1 a ( 3 ) b + 4 × 3 × 2 × 1 ( 2 × 1 ) ⋅ ( 2 × 1 ) a 2 b 2 + 4 × 3 × 2 × 1 1 ⋅ ( 3 × 2 × 1 ) a b 3 + b 4 = a 4 + 4 a 3 b + 6 a 2 b 2 + 4 a b 3 + b 4 Therefore, the expansion of ( a + b ) = 4 a 4 + 4 a 3 b + 6 a 2 b 2 + 4 a b 3 + b 4 For finding coefficients of expansion, use pascal triangle . First choose n =0, put in binomial theorem, So, ( a + b ) = 0 1 Second choose n = 1 , put in binomial theorem, Therefore, ( a + b ) = 1 ( a + b ) Now, the coefficient of a and b are 1 and 1 respectively. Third choose n = 2 , put in binomial theorem, ( a + b ) = 2 2 ! 2 ! ⋅ 0 ! a 2 + 2 ! ( 2 − 1 ) ! ⋅ 1 ! a ( 2 − 1 ) b + 2 ! ( 2 − 2 ) ! ⋅ 0 ! a ( 2 − 2 ) b 2 = a 2 + 2 a b + b 2 Therefore, the coefficient of a, ab and b is 1,2,1 respectively

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ISBN: 9780073384511

Author: Julie Miller, Molly O'Neill, Nancy Hyde

Publisher: MCGRAW-HILL HIGHER EDUCATION

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Polygon with three sides, three angles, and three vertices. Based on the properties of each side, the types of triangles are scalene (triangle with three three different lengths and three different angles), isosceles (angle with two equal sides and two equal angles), and equilateral (three equal sides and three angles of 60°). The types of angles are acute (less than 90°); obtuse (greater than 90°); and right (90°).

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To calculate: The expansion of (a+b)4 and determine the coefficient of the expansion.

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Chapter 14 Solutions

Beginning and Intermediate Algebra

Ch. 14.1 - Prob. 1SP Ch. 14.1 - Prob. 2SP Ch. 14.1 - Evaluate the expressions. 3. 1! Ch. 14.1 - Prob. 4SP Ch. 14.1 - Evaluate. 5!4!1!Ch. 14.1 - Evaluate. 5!3!2!Ch. 14.1 - Write out the first three terms of ( x + y ) 5 .Ch. 14.1 - 8. Use the binomial theorem to expand . Ch. 14.1 - Use the binomial theorem to expand ( 2 a − 3 b 2 )...Ch. 14.1 - Find the fourth term of ( x + y ) 8 .

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The expansion of ( a + b ) 4 and determine the coefficient of the expansion.

Solution Summary: The author calculates the expansion of the expression (a+b)4 by using the binomial theorem.

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To calculate: The expansion of (a+b)4 and determine the coefficient of the expansion.

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Chapter 14 Solutions