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Algebra 2 Section 4-2

The document discusses expanding powers of binomials using Pascal's triangle and the binomial theorem. It provides examples of expanding (p+t)5 and (t-w)8. Pascal's triangle provides the coefficients, and the binomial theorem formula is given as (a + b)n = Σk=0n (nCk * ak * bk), where the powers of the first term decrease and the second term increase in each term and sum to n.

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Section 4-2 Powers of Binomials
Essential Questions • How do you use Pascal’s Triangle to expand powers of binomials? • How do you use the Binomial Theorem to expand powers of binomials?
Vocabulary 1. Pascal’s Triangle:
Vocabulary 1. Pascal’s Triangle: A pattern of numbers that can be used to determine the coefficients to expand a binomial (a + b)n
Vocabulary 1. Pascal’s Triangle: A pattern of numbers that can be used to determine the coefficients to expand a binomial (a + b)n 1
Vocabulary 1. Pascal’s Triangle: A pattern of numbers that can be used to determine the coefficients to expand a binomial (a + b)n 1 1 1
Vocabulary 1. Pascal’s Triangle: A pattern of numbers that can be used to determine the coefficients to expand a binomial (a + b)n 1 1 1 1 12
Vocabulary 1. Pascal’s Triangle: A pattern of numbers that can be used to determine the coefficients to expand a binomial (a + b)n 1 1 1 1 12 3 31 1
Vocabulary 1. Pascal’s Triangle: A pattern of numbers that can be used to determine the coefficients to expand a binomial (a + b)n 1 1 1 1 12 3 31 1 1 4 4 16
Vocabulary 1. Pascal’s Triangle: A pattern of numbers that can be used to determine the coefficients to expand a binomial (a + b)n 1 1 1 1 12 3 31 1 1 4 4 16 10 105 51 1
Example 1 Expand (p +t)5
Example 1 Expand (p +t)5 (p +t)(p +t)(p +t)(p +t)(p +t)
Example 1 Expand (p +t)5 (p +t)(p +t)(p +t)(p +t)(p +t) (p2 + 2pt +t2 )(p2 + 2pt +t2 )(p +t )
Example 1 Expand (p +t)5 (p +t)(p +t)(p +t)(p +t)(p +t) (p2 + 2pt +t2 )(p2 + 2pt +t2 )(p +t ) (p4 + 2p3 t + p2 t2 + 2p3 t + 4p2 t2 + 2pt 3 + p2 t2 + 2pt 3 +t 4 )(p +t)
Example 1 Expand (p +t)5 (p +t)(p +t)(p +t)(p +t)(p +t) (p2 + 2pt +t2 )(p2 + 2pt +t2 )(p +t ) (p4 + 2p3 t + p2 t2 + 2p3 t + 4p2 t2 + 2pt 3 + p2 t2 + 2pt 3 +t 4 )(p +t) (p4 + 4p3 t + 6p2 t2 + 4pt 3 +t 4 )(p +t )
Example 1 Expand (p +t)5 (p +t)(p +t)(p +t)(p +t)(p +t) (p2 + 2pt +t2 )(p2 + 2pt +t2 )(p +t ) (p4 + 2p3 t + p2 t2 + 2p3 t + 4p2 t2 + 2pt 3 + p2 t2 + 2pt 3 +t 4 )(p +t) (p4 + 4p3 t + 6p2 t2 + 4pt 3 +t 4 )(p +t ) p5 + 4p4 t + 6p3 t2 + 4p2 t 3 + pt 4 + p4 t + 4p3 t2 + 6p2 t 3 + 4pt 4 +t5
Example 1 Expand (p +t)5 (p +t)(p +t)(p +t)(p +t)(p +t) (p2 + 2pt +t2 )(p2 + 2pt +t2 )(p +t ) (p4 + 2p3 t + p2 t2 + 2p3 t + 4p2 t2 + 2pt 3 + p2 t2 + 2pt 3 +t 4 )(p +t) (p4 + 4p3 t + 6p2 t2 + 4pt 3 +t 4 )(p +t ) p5 + 4p4 t + 6p3 t2 + 4p2 t 3 + pt 4 + p4 t + 4p3 t2 + 6p2 t 3 + 4pt 4 +t5 p5 + 5p4 t +10p3 t2 +10p2 t 3 + 5pt 4 +t5
Binomial Theorem (a + b)n = nC0an b0 + nC1an−1 b1 + nC2an−2 b2 +...+ nCna0 bn When expanding any binomial, we can use the pattern
Binomial Theorem (a + b)n = nC0an b0 + nC1an−1 b1 + nC2an−2 b2 +...+ nCna0 bn When expanding any binomial, we can use the pattern a + b( )n = n! k!(n − k )! an−k bk k =0 n ∑
Binomial Theorem (a + b)n = nC0an b0 + nC1an−1 b1 + nC2an−2 b2 +...+ nCna0 bn When expanding any binomial, we can use the pattern This uses Pascal’s Triangle! a + b( )n = n! k!(n − k )! an−k bk k =0 n ∑
Binomial Theorem (a + b)n = nC0an b0 + nC1an−1 b1 + nC2an−2 b2 +...+ nCna0 bn When expanding any binomial, we can use the pattern This uses Pascal’s Triangle! The powers of the first part count down. a + b( )n = n! k!(n − k )! an−k bk k =0 n ∑
Binomial Theorem (a + b)n = nC0an b0 + nC1an−1 b1 + nC2an−2 b2 +...+ nCna0 bn When expanding any binomial, we can use the pattern This uses Pascal’s Triangle! The powers of the first part count down. The powers of the second part count up. a + b( )n = n! k!(n − k )! an−k bk k =0 n ∑
Binomial Theorem (a + b)n = nC0an b0 + nC1an−1 b1 + nC2an−2 b2 +...+ nCna0 bn When expanding any binomial, we can use the pattern This uses Pascal’s Triangle! The powers of the first part count down. The powers of the second part count up. The powers within the term must add up to n. a + b( )n = n! k!(n − k )! an−k bk k =0 n ∑
Example 2 Expand (t −w)8 Binomial Theorem
Example 2 Expand (t −w)8 Binomial Theorem (t −w)8 = 8C0t8 (−w)0
Example 2 Expand (t −w)8 Binomial Theorem (t −w)8 = 8C0t8 (−w)0 +8C1t7 (−w)1
Example 2 Expand (t −w)8 Binomial Theorem (t −w)8 = 8C0t8 (−w)0 +8C1t7 (−w)1 +8C2t6 (−w)2
Example 2 Expand (t −w)8 Binomial Theorem (t −w)8 = 8C0t8 (−w)0 +8C1t7 (−w)1 +8C2t6 (−w)2 +8C3t5 (−w)3
Example 2 Expand (t −w)8 Binomial Theorem (t −w)8 = 8C0t8 (−w)0 +8C1t7 (−w)1 +8C2t6 (−w)2 +8C3t5 (−w)3 +8C4t 4 (−w)4
Example 2 Expand (t −w)8 Binomial Theorem (t −w)8 = 8C0t8 (−w)0 +8C1t7 (−w)1 +8C2t6 (−w)2 +8C3t5 (−w)3 +8C4t 4 (−w)4 +8C5t 3 (−w)5
Example 2 Expand (t −w)8 Binomial Theorem (t −w)8 = 8C0t8 (−w)0 +8C1t7 (−w)1 +8C2t6 (−w)2 +8C3t5 (−w)3 +8C4t 4 (−w)4 +8C5t 3 (−w)5 +8C6t2 (−w)6
Example 2 Expand (t −w)8 Binomial Theorem (t −w)8 = 8C0t8 (−w)0 +8C1t7 (−w)1 +8C2t6 (−w)2 +8C3t5 (−w)3 +8C4t 4 (−w)4 +8C5t 3 (−w)5 +8C6t2 (−w)6 +8C7t1 (−w)7
Example 2 Expand (t −w)8 Binomial Theorem (t −w)8 = 8C0t8 (−w)0 +8C1t7 (−w)1 +8C2t6 (−w)2 +8C3t5 (−w)3 +8C4t 4 (−w)4 +8C5t 3 (−w)5 +8C6t2 (−w)6 +8C7t1 (−w)7 +8C8t0 (−w)8
Example 2 Expand (t −w)8 Binomial Theorem
Example 2 Expand (t −w)8 Binomial Theorem (t −w)8 = t8
Example 2 Expand (t −w)8 Binomial Theorem (t −w)8 = t8 − 8! 1!7! t7 w1
Example 2 Expand (t −w)8 Binomial Theorem (t −w)8 = t8 − 8! 1!7! t7 w1 + 8! 2!6! t6 w2
Example 2 Expand (t −w)8 Binomial Theorem (t −w)8 = t8 − 8! 1!7! t7 w1 + 8! 2!6! t6 w2 − 8! 3!5! t5 w 3
Example 2 Expand (t −w)8 Binomial Theorem (t −w)8 = t8 − 8! 1!7! t7 w1 + 8! 2!6! t6 w2 − 8! 3!5! t5 w 3 + 8! 4!4! t 4 w 4
Example 2 Expand (t −w)8 Binomial Theorem (t −w)8 = t8 − 8! 1!7! t7 w1 + 8! 2!6! t6 w2 − 8! 3!5! t5 w 3 + 8! 4!4! t 4 w 4 − 8! 5!3! t 3 w5
Example 2 Expand (t −w)8 Binomial Theorem (t −w)8 = t8 − 8! 1!7! t7 w1 + 8! 2!6! t6 w2 − 8! 3!5! t5 w 3 + 8! 4!4! t 4 w 4 − 8! 5!3! t 3 w5 + 8! 6!2! t2 w6
Example 2 Expand (t −w)8 Binomial Theorem (t −w)8 = t8 − 8! 1!7! t7 w1 + 8! 2!6! t6 w2 − 8! 3!5! t5 w 3 + 8! 4!4! t 4 w 4 − 8! 5!3! t 3 w5 + 8! 6!2! t2 w6 − 8! 7!1! tw7
Example 2 Expand (t −w)8 Binomial Theorem (t −w)8 = t8 − 8! 1!7! t7 w1 + 8! 2!6! t6 w2 − 8! 3!5! t5 w 3 + 8! 4!4! t 4 w 4 − 8! 5!3! t 3 w5 + 8! 6!2! t2 w6 − 8! 7!1! tw7 +w8
Example 2 Expand (t −w)8 Binomial Theorem
Example 2 Expand (t −w)8 Binomial Theorem (t −w)8 = t8
Example 2 Expand (t −w)8 Binomial Theorem (t −w)8 = t8 −8t7 w
Example 2 Expand (t −w)8 Binomial Theorem (t −w)8 = t8 −8t7 w +28t6 w2
Example 2 Expand (t −w)8 Binomial Theorem (t −w)8 = t8 −8t7 w +28t6 w2 −56t5 w 3
Example 2 Expand (t −w)8 Binomial Theorem (t −w)8 = t8 −8t7 w +28t6 w2 −56t5 w 3 +70t 4 w 4
Example 2 Expand (t −w)8 Binomial Theorem (t −w)8 = t8 −8t7 w +28t6 w2 −56t5 w 3 +70t 4 w 4 −56t 3 w5
Example 2 Expand (t −w)8 Binomial Theorem (t −w)8 = t8 −8t7 w +28t6 w2 −56t5 w 3 +70t 4 w 4 −56t 3 w5 +28t2 w6
Example 2 Expand (t −w)8 Binomial Theorem (t −w)8 = t8 −8t7 w +28t6 w2 −56t5 w 3 +70t 4 w 4 −56t 3 w5 +28t2 w6 −8tw7
Example 2 Expand (t −w)8 Binomial Theorem (t −w)8 = t8 −8t7 w +28t6 w2 −56t5 w 3 +70t 4 w 4 −56t 3 w5 +28t2 w6 −8tw7 +w8
Example 2 Expand (t −w)8 Binomial Theorem (t −w)8 = t8 −8t7 w +28t6 w2 −56t5 w 3 +70t 4 w 4 −56t 3 w5 +28t2 w6 −8tw7 +w8 (t −w)8 = t8 − 8t7 w + 28t6 w2 − 56t5 w 3 + 70t 4 w 4 − 56t 3 w5 + 28t2 w6 − 8tw7 +w8
Example 2 Expand (t −w)8 Pascal’s Triangle
Example 2 Expand (t −w)8 Pascal’s Triangle Find the ninth row of the triangle for the coefficients of each term, make count down the exponents from the first part of your binomial and count up the exponents from the second term.
Example 2 Expand (t −w)8 Pascal’s Triangle (t −w)8 = t8 − 8t7 w + 28t6 w2 − 56t5 w 3 + 70t 4 w 4 − 56t 3 w5 + 28t2 w6 − 8tw7 +w8 Find the ninth row of the triangle for the coefficients of each term, make count down the exponents from the first part of your binomial and count up the exponents from the second term.
Example 3 Expand (a + 3b)4
Example 3 Expand (a + 3b)4 (a + 3b)4 = a4 + 4a3 (3b)1 + 6a2 (3b)2 + 4a(3b)3 + (3b)4
Example 3 Expand (a + 3b)4 (a + 3b)4 = a4 + 4a3 (3b)1 + 6a2 (3b)2 + 4a(3b)3 + (3b)4 (a + 3b)4 = a4 + 4a3 i 3b + 6a2 i 9b2 + 4a i 27b3 + 81b4
Example 3 Expand (a + 3b)4 (a + 3b)4 = a4 + 4a3 (3b)1 + 6a2 (3b)2 + 4a(3b)3 + (3b)4 (a + 3b)4 = a4 + 4a3 i 3b + 6a2 i 9b2 + 4a i 27b3 + 81b4 (a + 3b)4 = a4 +12a3 b + 54a2 b2 +108ab3 + 81b4
Example 3 Find the third term of (3x − y )4
Example 3 Find the third term of (3x − y )4 a + b( )n = n! k!(n − k )! an−k bk k =0 n ∑
Example 3 Find the third term of (3x − y )4 a + b( )n = n! k!(n − k )! an−k bk k =0 n ∑ 3x − y( )4 = 4! k!(4 − k )! (3x )4−k (−y )k k =0 4 ∑
Example 3 Find the third term of (3x − y )4 a + b( )n = n! k!(n − k )! an−k bk k =0 n ∑ 3x − y( )4 = 4! k!(4 − k )! (3x )4−k (−y )k k =0 4 ∑ 4! 2!(4 − 2)! (3x )4−2 (−y )2
Example 3 Find the third term of (3x − y )4 a + b( )n = n! k!(n − k )! an−k bk k =0 n ∑ 3x − y( )4 = 4! k!(4 − k )! (3x )4−k (−y )k k =0 4 ∑ 4! 2!(4 − 2)! (3x )4−2 (−y )2 4! 2!2! (3x )2 (−y )2
Example 3 Find the third term of (3x − y )4
Example 3 Find the third term of (3x − y )4 4! 2!2! (3x )2 (−y )2
Example 3 Find the third term of (3x − y )4 4! 2!2! (3x )2 (−y )2 6 i 9x 2 y 2
Example 3 Find the third term of (3x − y )4 4! 2!2! (3x )2 (−y )2 6 i 9x 2 y 2 54x 2 y 2
Example 3 Find the third term of (3x − y )4
Example 3 Find the third term of (3x − y )4 1
Example 3 Find the third term of (3x − y )4 1 1 1
Example 3 Find the third term of (3x − y )4 1 1 1 1 12
Example 3 Find the third term of (3x − y )4 1 1 1 1 12 3 31 1
Example 3 Find the third term of (3x − y )4 1 1 1 1 12 3 31 1 1 4 4 16
Example 3 Find the third term of (3x − y )4 1 1 1 1 12 3 31 1 1 4 4 16
Example 3 Find the third term of (3x − y )4 1 1 1 1 12 3 31 1 1 4 4 16 6(3x )2 (−y )2
Example 3 Find the third term of (3x − y )4 1 1 1 1 12 3 31 1 1 4 4 16 6(3x )2 (−y )2 6 i 9x 2 y 2
Example 3 Find the third term of (3x − y )4 1 1 1 1 12 3 31 1 1 4 4 16 6(3x )2 (−y )2 6 i 9x 2 y 2 54x 2 y 2

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Algebra 2 Section 4-2

  • 1.
  • 2.
    Essential Questions • Howdo you use Pascal’s Triangle to expand powers of binomials? • How do you use the Binomial Theorem to expand powers of binomials?
  • 3.
  • 4.
    Vocabulary 1. Pascal’s Triangle:A pattern of numbers that can be used to determine the coefficients to expand a binomial (a + b)n
  • 5.
    Vocabulary 1. Pascal’s Triangle:A pattern of numbers that can be used to determine the coefficients to expand a binomial (a + b)n 1
  • 6.
    Vocabulary 1. Pascal’s Triangle:A pattern of numbers that can be used to determine the coefficients to expand a binomial (a + b)n 1 1 1
  • 7.
    Vocabulary 1. Pascal’s Triangle:A pattern of numbers that can be used to determine the coefficients to expand a binomial (a + b)n 1 1 1 1 12
  • 8.
    Vocabulary 1. Pascal’s Triangle:A pattern of numbers that can be used to determine the coefficients to expand a binomial (a + b)n 1 1 1 1 12 3 31 1
  • 9.
    Vocabulary 1. Pascal’s Triangle:A pattern of numbers that can be used to determine the coefficients to expand a binomial (a + b)n 1 1 1 1 12 3 31 1 1 4 4 16
  • 10.
    Vocabulary 1. Pascal’s Triangle:A pattern of numbers that can be used to determine the coefficients to expand a binomial (a + b)n 1 1 1 1 12 3 31 1 1 4 4 16 10 105 51 1
  • 11.
  • 12.
    Example 1 Expand (p+t)5 (p +t)(p +t)(p +t)(p +t)(p +t)
  • 13.
    Example 1 Expand (p+t)5 (p +t)(p +t)(p +t)(p +t)(p +t) (p2 + 2pt +t2 )(p2 + 2pt +t2 )(p +t )
  • 14.
    Example 1 Expand (p+t)5 (p +t)(p +t)(p +t)(p +t)(p +t) (p2 + 2pt +t2 )(p2 + 2pt +t2 )(p +t ) (p4 + 2p3 t + p2 t2 + 2p3 t + 4p2 t2 + 2pt 3 + p2 t2 + 2pt 3 +t 4 )(p +t)
  • 15.
    Example 1 Expand (p+t)5 (p +t)(p +t)(p +t)(p +t)(p +t) (p2 + 2pt +t2 )(p2 + 2pt +t2 )(p +t ) (p4 + 2p3 t + p2 t2 + 2p3 t + 4p2 t2 + 2pt 3 + p2 t2 + 2pt 3 +t 4 )(p +t) (p4 + 4p3 t + 6p2 t2 + 4pt 3 +t 4 )(p +t )
  • 16.
    Example 1 Expand (p+t)5 (p +t)(p +t)(p +t)(p +t)(p +t) (p2 + 2pt +t2 )(p2 + 2pt +t2 )(p +t ) (p4 + 2p3 t + p2 t2 + 2p3 t + 4p2 t2 + 2pt 3 + p2 t2 + 2pt 3 +t 4 )(p +t) (p4 + 4p3 t + 6p2 t2 + 4pt 3 +t 4 )(p +t ) p5 + 4p4 t + 6p3 t2 + 4p2 t 3 + pt 4 + p4 t + 4p3 t2 + 6p2 t 3 + 4pt 4 +t5
  • 17.
    Example 1 Expand (p+t)5 (p +t)(p +t)(p +t)(p +t)(p +t) (p2 + 2pt +t2 )(p2 + 2pt +t2 )(p +t ) (p4 + 2p3 t + p2 t2 + 2p3 t + 4p2 t2 + 2pt 3 + p2 t2 + 2pt 3 +t 4 )(p +t) (p4 + 4p3 t + 6p2 t2 + 4pt 3 +t 4 )(p +t ) p5 + 4p4 t + 6p3 t2 + 4p2 t 3 + pt 4 + p4 t + 4p3 t2 + 6p2 t 3 + 4pt 4 +t5 p5 + 5p4 t +10p3 t2 +10p2 t 3 + 5pt 4 +t5
  • 18.
    Binomial Theorem (a +b)n = nC0an b0 + nC1an−1 b1 + nC2an−2 b2 +...+ nCna0 bn When expanding any binomial, we can use the pattern
  • 19.
    Binomial Theorem (a +b)n = nC0an b0 + nC1an−1 b1 + nC2an−2 b2 +...+ nCna0 bn When expanding any binomial, we can use the pattern a + b( )n = n! k!(n − k )! an−k bk k =0 n ∑
  • 20.
    Binomial Theorem (a +b)n = nC0an b0 + nC1an−1 b1 + nC2an−2 b2 +...+ nCna0 bn When expanding any binomial, we can use the pattern This uses Pascal’s Triangle! a + b( )n = n! k!(n − k )! an−k bk k =0 n ∑
  • 21.
    Binomial Theorem (a +b)n = nC0an b0 + nC1an−1 b1 + nC2an−2 b2 +...+ nCna0 bn When expanding any binomial, we can use the pattern This uses Pascal’s Triangle! The powers of the first part count down. a + b( )n = n! k!(n − k )! an−k bk k =0 n ∑
  • 22.
    Binomial Theorem (a +b)n = nC0an b0 + nC1an−1 b1 + nC2an−2 b2 +...+ nCna0 bn When expanding any binomial, we can use the pattern This uses Pascal’s Triangle! The powers of the first part count down. The powers of the second part count up. a + b( )n = n! k!(n − k )! an−k bk k =0 n ∑
  • 23.
    Binomial Theorem (a +b)n = nC0an b0 + nC1an−1 b1 + nC2an−2 b2 +...+ nCna0 bn When expanding any binomial, we can use the pattern This uses Pascal’s Triangle! The powers of the first part count down. The powers of the second part count up. The powers within the term must add up to n. a + b( )n = n! k!(n − k )! an−k bk k =0 n ∑
  • 24.
    Example 2 Expand (t−w)8 Binomial Theorem
  • 25.
    Example 2 Expand (t−w)8 Binomial Theorem (t −w)8 = 8C0t8 (−w)0
  • 26.
    Example 2 Expand (t−w)8 Binomial Theorem (t −w)8 = 8C0t8 (−w)0 +8C1t7 (−w)1
  • 27.
    Example 2 Expand (t−w)8 Binomial Theorem (t −w)8 = 8C0t8 (−w)0 +8C1t7 (−w)1 +8C2t6 (−w)2
  • 28.
    Example 2 Expand (t−w)8 Binomial Theorem (t −w)8 = 8C0t8 (−w)0 +8C1t7 (−w)1 +8C2t6 (−w)2 +8C3t5 (−w)3
  • 29.
    Example 2 Expand (t−w)8 Binomial Theorem (t −w)8 = 8C0t8 (−w)0 +8C1t7 (−w)1 +8C2t6 (−w)2 +8C3t5 (−w)3 +8C4t 4 (−w)4
  • 30.
    Example 2 Expand (t−w)8 Binomial Theorem (t −w)8 = 8C0t8 (−w)0 +8C1t7 (−w)1 +8C2t6 (−w)2 +8C3t5 (−w)3 +8C4t 4 (−w)4 +8C5t 3 (−w)5
  • 31.
    Example 2 Expand (t−w)8 Binomial Theorem (t −w)8 = 8C0t8 (−w)0 +8C1t7 (−w)1 +8C2t6 (−w)2 +8C3t5 (−w)3 +8C4t 4 (−w)4 +8C5t 3 (−w)5 +8C6t2 (−w)6
  • 32.
    Example 2 Expand (t−w)8 Binomial Theorem (t −w)8 = 8C0t8 (−w)0 +8C1t7 (−w)1 +8C2t6 (−w)2 +8C3t5 (−w)3 +8C4t 4 (−w)4 +8C5t 3 (−w)5 +8C6t2 (−w)6 +8C7t1 (−w)7
  • 33.
    Example 2 Expand (t−w)8 Binomial Theorem (t −w)8 = 8C0t8 (−w)0 +8C1t7 (−w)1 +8C2t6 (−w)2 +8C3t5 (−w)3 +8C4t 4 (−w)4 +8C5t 3 (−w)5 +8C6t2 (−w)6 +8C7t1 (−w)7 +8C8t0 (−w)8
  • 34.
    Example 2 Expand (t−w)8 Binomial Theorem
  • 35.
    Example 2 Expand (t−w)8 Binomial Theorem (t −w)8 = t8
  • 36.
    Example 2 Expand (t−w)8 Binomial Theorem (t −w)8 = t8 − 8! 1!7! t7 w1
  • 37.
    Example 2 Expand (t−w)8 Binomial Theorem (t −w)8 = t8 − 8! 1!7! t7 w1 + 8! 2!6! t6 w2
  • 38.
    Example 2 Expand (t−w)8 Binomial Theorem (t −w)8 = t8 − 8! 1!7! t7 w1 + 8! 2!6! t6 w2 − 8! 3!5! t5 w 3
  • 39.
    Example 2 Expand (t−w)8 Binomial Theorem (t −w)8 = t8 − 8! 1!7! t7 w1 + 8! 2!6! t6 w2 − 8! 3!5! t5 w 3 + 8! 4!4! t 4 w 4
  • 40.
    Example 2 Expand (t−w)8 Binomial Theorem (t −w)8 = t8 − 8! 1!7! t7 w1 + 8! 2!6! t6 w2 − 8! 3!5! t5 w 3 + 8! 4!4! t 4 w 4 − 8! 5!3! t 3 w5
  • 41.
    Example 2 Expand (t−w)8 Binomial Theorem (t −w)8 = t8 − 8! 1!7! t7 w1 + 8! 2!6! t6 w2 − 8! 3!5! t5 w 3 + 8! 4!4! t 4 w 4 − 8! 5!3! t 3 w5 + 8! 6!2! t2 w6
  • 42.
    Example 2 Expand (t−w)8 Binomial Theorem (t −w)8 = t8 − 8! 1!7! t7 w1 + 8! 2!6! t6 w2 − 8! 3!5! t5 w 3 + 8! 4!4! t 4 w 4 − 8! 5!3! t 3 w5 + 8! 6!2! t2 w6 − 8! 7!1! tw7
  • 43.
    Example 2 Expand (t−w)8 Binomial Theorem (t −w)8 = t8 − 8! 1!7! t7 w1 + 8! 2!6! t6 w2 − 8! 3!5! t5 w 3 + 8! 4!4! t 4 w 4 − 8! 5!3! t 3 w5 + 8! 6!2! t2 w6 − 8! 7!1! tw7 +w8
  • 44.
    Example 2 Expand (t−w)8 Binomial Theorem
  • 45.
    Example 2 Expand (t−w)8 Binomial Theorem (t −w)8 = t8
  • 46.
    Example 2 Expand (t−w)8 Binomial Theorem (t −w)8 = t8 −8t7 w
  • 47.
    Example 2 Expand (t−w)8 Binomial Theorem (t −w)8 = t8 −8t7 w +28t6 w2
  • 48.
    Example 2 Expand (t−w)8 Binomial Theorem (t −w)8 = t8 −8t7 w +28t6 w2 −56t5 w 3
  • 49.
    Example 2 Expand (t−w)8 Binomial Theorem (t −w)8 = t8 −8t7 w +28t6 w2 −56t5 w 3 +70t 4 w 4
  • 50.
    Example 2 Expand (t−w)8 Binomial Theorem (t −w)8 = t8 −8t7 w +28t6 w2 −56t5 w 3 +70t 4 w 4 −56t 3 w5
  • 51.
    Example 2 Expand (t−w)8 Binomial Theorem (t −w)8 = t8 −8t7 w +28t6 w2 −56t5 w 3 +70t 4 w 4 −56t 3 w5 +28t2 w6
  • 52.
    Example 2 Expand (t−w)8 Binomial Theorem (t −w)8 = t8 −8t7 w +28t6 w2 −56t5 w 3 +70t 4 w 4 −56t 3 w5 +28t2 w6 −8tw7
  • 53.
    Example 2 Expand (t−w)8 Binomial Theorem (t −w)8 = t8 −8t7 w +28t6 w2 −56t5 w 3 +70t 4 w 4 −56t 3 w5 +28t2 w6 −8tw7 +w8
  • 54.
    Example 2 Expand (t−w)8 Binomial Theorem (t −w)8 = t8 −8t7 w +28t6 w2 −56t5 w 3 +70t 4 w 4 −56t 3 w5 +28t2 w6 −8tw7 +w8 (t −w)8 = t8 − 8t7 w + 28t6 w2 − 56t5 w 3 + 70t 4 w 4 − 56t 3 w5 + 28t2 w6 − 8tw7 +w8
  • 55.
    Example 2 Expand (t−w)8 Pascal’s Triangle
  • 56.
    Example 2 Expand (t−w)8 Pascal’s Triangle Find the ninth row of the triangle for the coefficients of each term, make count down the exponents from the first part of your binomial and count up the exponents from the second term.
  • 57.
    Example 2 Expand (t−w)8 Pascal’s Triangle (t −w)8 = t8 − 8t7 w + 28t6 w2 − 56t5 w 3 + 70t 4 w 4 − 56t 3 w5 + 28t2 w6 − 8tw7 +w8 Find the ninth row of the triangle for the coefficients of each term, make count down the exponents from the first part of your binomial and count up the exponents from the second term.
  • 58.
  • 59.
    Example 3 Expand (a+ 3b)4 (a + 3b)4 = a4 + 4a3 (3b)1 + 6a2 (3b)2 + 4a(3b)3 + (3b)4
  • 60.
    Example 3 Expand (a+ 3b)4 (a + 3b)4 = a4 + 4a3 (3b)1 + 6a2 (3b)2 + 4a(3b)3 + (3b)4 (a + 3b)4 = a4 + 4a3 i 3b + 6a2 i 9b2 + 4a i 27b3 + 81b4
  • 61.
    Example 3 Expand (a+ 3b)4 (a + 3b)4 = a4 + 4a3 (3b)1 + 6a2 (3b)2 + 4a(3b)3 + (3b)4 (a + 3b)4 = a4 + 4a3 i 3b + 6a2 i 9b2 + 4a i 27b3 + 81b4 (a + 3b)4 = a4 +12a3 b + 54a2 b2 +108ab3 + 81b4
  • 62.
    Example 3 Find thethird term of (3x − y )4
  • 63.
    Example 3 Find thethird term of (3x − y )4 a + b( )n = n! k!(n − k )! an−k bk k =0 n ∑
  • 64.
    Example 3 Find thethird term of (3x − y )4 a + b( )n = n! k!(n − k )! an−k bk k =0 n ∑ 3x − y( )4 = 4! k!(4 − k )! (3x )4−k (−y )k k =0 4 ∑
  • 65.
    Example 3 Find thethird term of (3x − y )4 a + b( )n = n! k!(n − k )! an−k bk k =0 n ∑ 3x − y( )4 = 4! k!(4 − k )! (3x )4−k (−y )k k =0 4 ∑ 4! 2!(4 − 2)! (3x )4−2 (−y )2
  • 66.
    Example 3 Find thethird term of (3x − y )4 a + b( )n = n! k!(n − k )! an−k bk k =0 n ∑ 3x − y( )4 = 4! k!(4 − k )! (3x )4−k (−y )k k =0 4 ∑ 4! 2!(4 − 2)! (3x )4−2 (−y )2 4! 2!2! (3x )2 (−y )2
  • 67.
    Example 3 Find thethird term of (3x − y )4
  • 68.
    Example 3 Find thethird term of (3x − y )4 4! 2!2! (3x )2 (−y )2
  • 69.
    Example 3 Find thethird term of (3x − y )4 4! 2!2! (3x )2 (−y )2 6 i 9x 2 y 2
  • 70.
    Example 3 Find thethird term of (3x − y )4 4! 2!2! (3x )2 (−y )2 6 i 9x 2 y 2 54x 2 y 2
  • 71.
    Example 3 Find thethird term of (3x − y )4
  • 72.
    Example 3 Find thethird term of (3x − y )4 1
  • 73.
    Example 3 Find thethird term of (3x − y )4 1 1 1
  • 74.
    Example 3 Find thethird term of (3x − y )4 1 1 1 1 12
  • 75.
    Example 3 Find thethird term of (3x − y )4 1 1 1 1 12 3 31 1
  • 76.
    Example 3 Find thethird term of (3x − y )4 1 1 1 1 12 3 31 1 1 4 4 16
  • 77.
    Example 3 Find thethird term of (3x − y )4 1 1 1 1 12 3 31 1 1 4 4 16
  • 78.
    Example 3 Find thethird term of (3x − y )4 1 1 1 1 12 3 31 1 1 4 4 16 6(3x )2 (−y )2
  • 79.
    Example 3 Find thethird term of (3x − y )4 1 1 1 1 12 3 31 1 1 4 4 16 6(3x )2 (−y )2 6 i 9x 2 y 2
  • 80.
    Example 3 Find thethird term of (3x − y )4 1 1 1 1 12 3 31 1 1 4 4 16 6(3x )2 (−y )2 6 i 9x 2 y 2 54x 2 y 2