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ALGEBRA II HONORS/GIFTED @

ALGEBRA II HONORS/GIFTED @. SECTION 5-7 : THE BINOMIAL THEOREM. Work out the problem that corresponds to your group number. 9) What do you notice about the signs before each term?. (x + y) 0 (x – y) 0 (x + y) 1 (x – y) 1 (x + y) 2 (x – y) 2 (x + y) 3 8) (x – y) 3. = 1 = 1

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ALGEBRA II HONORS/GIFTED @

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  1. ALGEBRA II HONORS/GIFTED @ SECTION 5-7 : THE BINOMIAL THEOREM

  2. Work out the problem that corresponds to your group number. 9) What do you notice about the signs before each term? • (x + y)0 • (x – y)0 • (x + y)1 • (x – y)1 • (x + y)2 • (x – y)2 • (x + y)3 • 8) (x – y)3 = 1 = 1 = x + y = x – y = x2 + 2xy + y2 = x2 – 2xy + y2 = x3 + 3x2y + 3xy2 + y3 = x3 – 3x2y + 3xy2 – y3 10) And, those exponents. What about them? 11) Hey, what’s going on with the coefficients?

  3. 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1 1 7 21 35 35 21 7 1 1 8 28 56 70 56 28 8 1 This is Pascal’s Triangle. The triangle’s row are numbered beginning with row 0.

  4. Expand. a) How many terms are in the expansion? 9a) (x + y)4 ANSWER : 5 b) How are the signs of the terms arranged? + + + + + (1) (4) (6) (4) (1) (x)4 (x)3 (x)2 (x)1 (x)0 (y)0 (y)1 (y)2 (y)3 (y)4 ANSWER : all + c) Lay down the coefficients from Pascal’s Triangle. d) Place the terms of the binomial down noting their exponents. e) Simplify. = x4 + 4x3y + 6x2y2 + 4xy3 + y4

  5. f) (x – y)5 ANSWER : x5 - 5x4y + 10x3y2 - 10x2y3 + 5xy4 - y5 g) (x – y)6 ANSWER : x6 – 6x5y + 15x4y2 – 20x3y3 + 15x2y4 – 6xy5 + y6 Find the specified term. 10a) Third term of (x + 2y)6 ANSWER : 60x4y2 b) Fourth term of (2x2 – y)5 ANSWER : -40x4y3

  6. 11)a) Expand using Pascal’s Triangle : (x – 4)5 ANSWER : x5 – 20x4 + 160x3 – 640x2 + 1280x - 1024 b) Now, find : 5C0, 5C1, 5C2, 5C3, 5C4, and 5C5. ANSWERS : 1, 5, 10, 10, 5, 1 c) How do these answers compare with the coefficients found in Pascal’s Triangle? ANSWER : They are the same.

  7. 12) Expand. SOLUTION : +4C0 (3x)4(2)0 -4C1(3x)3(2)1 +4C2(3x)2(2)2 -4C3(3x)1(2)3 +4C4(3x)0(2)4 a) (3x – 2)4 = 81x4 – 216x3 + 216x2 – 96x + 16

  8. b) (4x + y2)3 ANSWER : 64x3 + 48x2y2 + 12xy4 + y6 c) (a + a-1)6 ANSWER : a6 + 6a4 + 15a2 + 20 + 15a-2 + 6a-4 + a-6 d) (1 – i)5 ANSWER : -4 + 4i

  9. 13) Find the specified term. Must add to 9. a) (x + y)9 ; y4 SOLUTION : 9C4 (x)5 (y)4 =126x5y4 Where does the “9” come from? These numbers are always the same. b) (2x – 3y)7 ; y5 ANSWER : 7C5(2x)2(-3y)5= -20412x2y5

  10. c) (a – 2b)8 ; a3 ANSWER : 8C5 (a)3 (-2b)5 = -1792a3b5 d) (a + b)n ; br ANSWER : nCr (a)n-r (b)r

  11. 14) Find the specified term. a) (m + n)20 ; 12th term SOLUTION : 20C11 (m)9 (n)11 = 167960m9n11 The choose number is always 1 less than the term number. The choose number is also known as the “magic number”. b) (r3 – 2s)9 ; 5th term ANSWER : 9C4 (r3)5 (-2s)4 = 2016r15s4

  12. c) (2x – y)12 ; middle term ANSWER : 12C6 (2x)6 (-y)6 = 59136x6y6 d) (a + 2b3)10 ; middle term ANSWER : 10C5 (a)5 (2b)5 = 8064a5b5

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