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Pascal’s Triangle

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Pascal’s Triangle

1. 1. A Tool for Finding the Power of Binomials
2. 2. <ul><li>What is It? </li></ul><ul><li>Basic Definition: </li></ul><ul><ul><li>A triangular pattern of numbers in which each number is equal to the sum of the two numbers immediately above it. </li></ul></ul><ul><li>Mathematic Definition: </li></ul><ul><ul><li>A geometric arrangement of the binomial coefficients in a triangle </li></ul></ul>
3. 3. <ul><li>Write the number 1 </li></ul><ul><li>Write two more 1’s underneath (forming a triangle) </li></ul><ul><li>Write two more 1’s underneath (to the left & right ) </li></ul><ul><li>Now add the two 1’s and put the sum underneath in the middle </li></ul><ul><li>Follow this pattern </li></ul>1 1 1 1 1 2 3 1 3 1 6 1 4 4 1 . . .
4. 4. <ul><li>Each row represents the coefficients of the power of binomials. </li></ul>(u+v) 0 = 1 (u+v) 1 = u + v (u+v) 2 = u 2 + 2uv + v 2 (u+v) 3 = u 3 + 3u 2 v + 3uv 2 + v 3 (u+v) 4 = u 4 + 4u 3 v + 6u 2 v 2 + 4uv 3 + v 4 NOTE: We do not write coefficients of 1.
5. 5. <ul><li>If the coefficients of “1” are included, we can see Pascal’s Triangle forming. </li></ul>(u+v) 0 = 1 (u+v) 1 = 1 u + 1 v (u+v) 2 = 1 u 2 + 2 uv + 1 v 2 (u+v) 3 = 1 u 3 + 3 u 2 v + 3 uv 2 + 1 v 3 (u+v) 4 = 1 u 4 + 4 u 3 v + 6 u 2 v 2 + 4 uv 3 + 1 v 4
6. 6. <ul><li>If we change the operation to subtraction, we rotate a “+” & “-” sign in the triangle </li></ul>(u - v) 0 = 1 (u - v) 1 = u - v (u - v) 2 = u 2 - 2uv + v 2 (u - v) 3 = u 3 - 3u 2 v + 3uv 2 - v 3 (u - v) 4 = u 4 - 4u 3 v + 6u 2 v 2 - 4uv 3 + v 4
7. 7. <ul><li>Examples </li></ul><ul><li>Expand the following: ( x + 5) 3 </li></ul>x 3 + 3( x 2 )(5) + 3( x )(5 2 ) + 5 3 x 3 + 15 x 2 + 75 x + 125
8. 8. <ul><li>Examples </li></ul><ul><li>Expand the following: ( x - 2) 4 </li></ul>x 4 – 4( x 3 )(2) + 6( x 2 )(2 2 ) – 4( x )(2 3 ) + 2 4 x 4 – 8 x 3 + 24 x 2 – 32 x + 16
9. 9. <ul><li>Examples </li></ul><ul><li>Expand the following: (2 x + 3) 3 </li></ul>(2 x ) 3 + 3(2 x ) 2 (3) + 3(2 x )(3) 2 + 3 3 8 x 3 + 36 x 2 + 54 x + 27
10. 10. <ul><li>Examples </li></ul><ul><li>Expand the following: (5x - 7) 2 </li></ul>(5 x ) 2 – 2(5 x )(7) + 7 3 25 x 2 - 70 x + 343