pascal’s triangle and the binomial theorem, then exam! 20.0 students know the binomial theorem and...
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Pascal’s Triangle and the Binomial Theorem, then Exam!
20.0 Students know the binomial theorem and use it to expand binomial expressions that are raised to positive
integer powers.
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Pascal’s Triangle and the Binomial Theorem
Objectives• Relate Pascal’s Triangle to
the terms of a Binomial Expansion– The Binomial Theorem
Key Words• Pascal’s triangle
– The arrangement of in a triangular pattern in which each row corresponds to a value of n. (pg 553, you have to see it to believe it!)
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Pascal’s Triangle
If you arrange the values of in a triangular pattern in which each row corresponds to a value of n, you get a pattern called Pascal’s triangle.
Turn to page 553.
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The Binomial Theorem
For any positive integer n, the expansion of is:
+
Note that each term has the form where r is an integer from 0 to n.
Examples:
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Example 1 Expand a Power of a Simple Binomial Sum
Coefficients: 1, 4, 6, 4, 1
Powers of a: a 4, a 3, a 2, a 1, a 0
Powers of b: b0, b1, b2, b3, b4
Expand ( )4. ba +
SOLUTION
In , the power is n 4. So, the coefficients of the terms are the numbers in the 4th row of Pascal’s Triangle.
=( )4ba +
= 1a 4b 0 + + + +( )4ba + 4a 3b 1 6a 2b 2 1a 0b 44a 1b 3
= a 4 + + + +4a 3b 6a 2b 2 4ab 3 b 4
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Example 2 Expand a Power of a Binomial Sum
SOLUTION
Use the binomial theorem with a x and b 5.= =
Expand ( )3. 5x +
= + + +( )x 3( )1 ( )1 ( )x 2( )3 ( )5 ( )x 1( )3 ( )25 ( )x 0( )1 ( )125
= +x 3 15x 2 75x 125+ +
5x( +
(
3 = 3C0x 350 + + +3C1x 251 3C2x 152
3C3x 053
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Example 3 Expand a Power of a Binomial Difference
First rewrite the difference as a sum:
SOLUTION
= 2x[ + 4(
( – y ]y2x( – ) 4
= =Then use the binomial theorem with a 2x and b –y.= =
Expand y2x( – )4.
4C3 4C2
4C4
= + +2x[ + 4(
( – y ]4C0
(
(2x 4 (
( – y 04C1
(
(2x 3 (
( – y 1
(
(2x 2 (
( – y 2 +
(
(2x 1 (
( – y 3 +
(
(2x 0 (
( – y 4
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Example 3 Expand a Power of a Binomial Difference
= + + +
+
( )16x 4( )1 ( )1 ( )8x 3( )4 ( )y– ( )4x 2( )6 ( )y 2
( )2x4 ( )– y 3 ( )1 ( )1 ( )y 4
= 16x 4 + + – +32x 3y 24x 2y 2 8xy 3 y 4
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CheckpointExpand a Power of a Binomial Sum or Difference
ANSWER a 5 + + + + +5a 4b 10a 3b 2 10a 2b 3 5ab 4 b 5
ANSWER x 4 + + + +8x 3 24x 2 32x 16
Expand the power of the binomial sum or difference.
1. ba( + )5
2. 2x( + )4
3. 53x( + )3
ANSWER 27x 3 + + +135x 2 225x 125
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CheckpointExpand a Power of a Binomial Sum or Difference
5. – nm( )4
6. t3s( – )3
Expand the power of the binomial sum or difference.
4. 4p( )3–
p 3ANSWER +– –12p 2 48p 64
m 4ANSWER +– – +4m 3n 6m 2n 2 4mn 3 n 4
27s 3 +– –27s 2t 9st 2 t 3ANSWER
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Conclusions
SummaryExit Slip:• How can you calculate the
coefficients of the terms of ?– Each term in the expansion of
has the form , where r is an integer from 0 to n.
Assignment• Pg 555
– #(2,6-13)
• Write the assignment down, you will work on it after the you finish the exam, early.
• Get ready for the exam.
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45 MINUTESNO TALKING – READ RUBRIC – READ DIRECTIONS – GOOD LUCK!
Exam on the Fundamental Counting Principle