How to Calculate (x+y)^n with Pascal's Triangle
On a blank piece of paper, draw up Pascal's triangle, with some space reserved to the right., On the right of each row of the Pascal's triangle, write (x+y). , Now, take the power of n, n+1, n+2... onto (x+y).,If you can see from Pascal's triangle...
Step-by-Step Guide
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Step 1: On a blank piece of paper
Draw the triangle up to at least 5 rows. -
Step 2: draw up Pascal's triangle
, In other terms, start with (x+y)0, then (x+y)1, (x+y)2.
Note down the powers.
N to the power of 0 is always
1.
That means (x+y)0 is
1.
Also, (x+y)1 is simply (x+y). ,,,, In mathematical terms, for each 1 you see, except the first line, it will represent either the coefficients of x or y.
It will always have the highest power used in your (x+y)N, which is N.
The numbers in a line in Pascal's triangle will refer to the coefficients of each term, with the amount of number in a line referring to the total amount of terms related to (x+y)N. ,, Example: (x+y)4 Since the power (n) = 4, we should have a look at the fifth (n+1)th row of the Pascal triangle.
Thus, the (n + 1 = 5)th row of the Pascal triangle is:1 4 6 4 1 Therefore, 1 4 6 4 1 represent the coefficients of the terms of x & y after expansion of (x+y)4.
The answer: x4+4x3y+6x2y2+4xy3+y4 -
Step 3: with some space reserved to the right.
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Step 4: On the right of each row of the Pascal's triangle
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Step 5: write (x+y).
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Step 6: take the power of n
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Step 7: n+2... onto (x+y).
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Step 8: If you can see from Pascal's triangle
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Step 9: the first 1 represents (x+y)
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Step 10: with 1 as 0.
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Step 11: The next row of the Pascal's triangle
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Step 12: 1) represents (x+y)1
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Step 13: with the first 1 as the coefficient of x
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Step 14: and the second as coefficient of y.
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Step 15: The next row of Pascal's triangle
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Step 16: 1) represents (x+y)2
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Step 17: with the first 1 as as the coefficient of x2
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Step 18: second as the coefficient of 2xy and third as coefficient of y2.
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Step 19: Confused?
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Step 20: Each x term power will decrease over the terms
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Step 21: like: x3
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Step 22: then x2
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Step 23: then x
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Step 24: and then 1: which represents NIL in this process.
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Step 25: Each y term power will increase over the terms
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Step 26: 1: which represents NIL in this process
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Step 27: then y2
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Step 28: then y3.
Detailed Guide
Draw the triangle up to at least 5 rows.
, In other terms, start with (x+y)0, then (x+y)1, (x+y)2.
Note down the powers.
N to the power of 0 is always
1.
That means (x+y)0 is
1.
Also, (x+y)1 is simply (x+y). ,,,, In mathematical terms, for each 1 you see, except the first line, it will represent either the coefficients of x or y.
It will always have the highest power used in your (x+y)N, which is N.
The numbers in a line in Pascal's triangle will refer to the coefficients of each term, with the amount of number in a line referring to the total amount of terms related to (x+y)N. ,, Example: (x+y)4 Since the power (n) = 4, we should have a look at the fifth (n+1)th row of the Pascal triangle.
Thus, the (n + 1 = 5)th row of the Pascal triangle is:1 4 6 4 1 Therefore, 1 4 6 4 1 represent the coefficients of the terms of x & y after expansion of (x+y)4.
The answer: x4+4x3y+6x2y2+4xy3+y4
About the Author
James Mitchell
Dedicated to helping readers learn new skills in crafts and beyond.
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