Binomial theorem describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the power (x + y)n into a sum involving terms of the form axbyc, where the exponents b and c are non-negative integers with b + c = n, and the coefficient a of each term is a specific positive integer depending on n and b. When an exponent is zero, the corresponding power is usually omitted from the term.
I'll give you a quick guide.
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Say you have to expand the binomial (x + y)
3.
Step 1: Write x terms from power 3 to 0.
x
3 + x
2 + x
1 + x
0
Step 2: Add y terms from power 0 to 3 after the x term (In reverse order).
x
3y0 + x
2y1 + x
1y2 + x
0y3
Step 3: Remove power 0 terms (x
0 = y
0 = 1). Also write power 1 terms without power sysmbol (x1 = x, y1 = y)
x
3 + x
2y + xy
2 + y
3
Step 4: Add coefficients (??????) according to Pascal Triangle for Power 3 (4th (3+1) raw) since we are expanding (x + y)
3.
1 3 3 1 are the values from Pascal Triangle for Power 3 (4th raw). These are called binomial coefficients.
1x
3 +
3x
2y +
3xy
2 +
1y
3
Finally we get:
(x + y)
3 = x
3 + 3x
2y + 3xy
2 + y
3
Let's evaluate (a + b)
5 quickly now.
Step 1and 2 together:
a
5b
0 + a
4b
1 + a
3b
2 + a
2b
3 + a
1b
4 + a
0b
5
Step 3:
a
5 + a
4b + a
3b
2 + a
2b
3 + ab
4 + b
5
Step 4:
Chose the binomial coefficients from Pascal triangle. We need the value for Power 5. So we look at the 6th raw.
1 5 10 10 5 1
1a
5 + 5a
4b + 10a
3b
2 + 10a
2b
3 + 5ab
4 + 1b
5
Finally we have,
(a + b)
5 = a
5 + 5a
4b + 10a
3b
2 + 10a
2b
3 + 5ab
4 + b
5
I guess you are now clear on the process.