# 7.7 Binomial Theorem

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```					MCR 3U0
Unit 7: Sequences, Series & Investments
Lesson 6: The Binomial Theorem
Page 1 of 3
Investigation: The Binomial Theorem Workbook Page 3
1 Expand and simplify the following expression fully.
a) (a + b ) 1
=a+b
b) (a + b ) 2
= a 2 + 2ab + b 2
c) (a + b ) 3
= (a + b ) (a 2 + 2ab + b 2 )
= a 3 + 2a 2b + ab 2 + a 2b + 2ab 2 + b 3
= a 3 + 3a 2b + 3ab 2 + b 3
d) (a + b ) 4
= (a + b )(a 3 + 3a 2b + 3ab 2 + b 3)
= a 4 + 3a 3b + 3a 2b 2 + ab 3 +
a 3b + 3a 2b 2 + 3ab 3 + b 4
= a 4 + 4a 3b + 6a 2b 2 + 4ab 3 + b 4
e) (a + b ) 5
= (a + b )(a 4 + 4a 3b + 6a 2b 2 + 4ab 3 + b 4 )
= a 5 + 4a 4b + 6a 3b 2 + 4a 2b 3 + ab 4 + a 4b + 4a 3b 2 + 6a 2b 3 + 4ab 4 + b 5
= a 5 + 5a 4b + 10a 3b 2 + 10a 2b 3 + 5ab 4 + b 5
2 Study the terms in each of these expansions. Describe how the degree of each
term relates to the exponent of the binomial.
The degree of each term in the polynomial is equal to the exponent on the
binomial.
3 Compare the terms in Pascal’s triangle to the expansions in Question 1.
The terms in Pascal’s triangle are the coefficients of the terms in the
polynomial. The coefficients are taken from the row that corresponds to the
exponent of the polynomial.
4 Predict the terms of the expansion of (a + b ) 6.
= a 6 + 6a 5 b + 15a 4 b 2 + 20 a 3 b 3 + 15a 2 b 4 + 6a b 5 + b 6
MCR 3U0
Unit 7: Sequences, Series & Investments
Lesson 6: The Binomial Theorem
Page 2 of 3
Recall: Pascal’s Triangle
1
11
121
1331
14641
1 5 10 10 5 1
1 6 15 20 15 6 1
1 7 21 35 35 21 7 1
Conclusions:
• The coefficients of the expansion correspond to the row of Pascal’s triangle
that
is the same as the exponent on the expansion. (Note: The first row of Pascal’s
triangle is Row 0.)
• The first term of the polynomial is the first term in the binomial raised to
the exponent of the binomial. That is, for
(a + b ) 6,
the first term is
a6
(Note: The second term of binomial has as exponent of 0.)
The first term of the polynomial is really.
a 6b 0
• The second term of the polynomial is the first term in the binomial raised to
one less than the exponent of the binomial multiplied with the second term of
the binomial to the exponent 1. That is, for
(a + b ) 6,
the second term is
a 6-1b 0+1
etc.
• The number of terms in the expansion is one more than the exponent of the
expansion.
MCR 3U0
Unit 7: Sequences, Series & Investments
Lesson 6: The Binomial Theorem
Page 3 of 3
Example 1: Use Pascal’s Triangle to expand the following expressions.
a) (x + 2 ) 4
Since the exponent is 4, we need to refer to the fourth row of Pascal’s
triangle to determine our coefficients.
= 1 (x ) 4 (2)0 + 4 (x ) 3 (2)1 + 6 (x ) 2 (2)2 + 4 (x )1 (2)3 + 1 (x )0 (2)4
= x 4 + 4 x 3 (2) + 6x 2 (4) + 4 x (8) + 16
= x 4 + 8x 3 + 24x 2 + 32 x + 16
b) (2x - 1 ) 4
Since the exponent is 4, we need to refer to the fourth row of Pascal’s
triangle to determine our coefficients.
= 1 (2x ) 4 (-1)0 + 4 (2x ) 3 (-1)1 + 6 (2x ) 2 (-1)2 + 4 (2x )1 (-1)3
+ 1 (2x )0 (-1)4
= 16x 4 + 4 (8x 3 )(-1) + 6(4x 2 )(1) + 4 (2x) (-1) + 1
= 16x 4 - 32x 3 + 24x 2 - 8 x + 1
c) (3x - 2y )5
Since the exponent is 5, we need to refer to the fifth row of Pascal’s
triangle to determine our coefficients.
= 1 (3x ) 5 (-2y )0 + 5 (3x ) 4 (-2y )1 + 10 (3x ) 3 (-2y )2
+ 10 (3x )2 (-2y )3 + 5 (3x )1 (-2y )4 + 1 (3x )0 (-2y ) 5
= 243x 5 + 5 (81x 4 )(-2y ) + 10(27x 3 )(4y 2 ) + 10 (9x 2) (-8y 3)
+ 5(3x)(16y 4 ) - 32y 5
= 243x 5 - 810x 4 y + 1080x 3y 2 - 720x 2y 3 + 240xy 4 - 32y 5

Homework: Exercise 6: The Binomial Theorem
Workbook Page 4
Investigate & Inquire Pages 465 - 466

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