BINOMIAL EXPANSION

BINOMIAL EXPANSIONS When a binomial of the form 𝑎 + 𝑏 is raised to a power,

the resulting polynomial can be thought of as series.

Suppose we expand several such powers and search

for a pattern:

𝑎 + 𝑏 0 1

𝑎 + 𝑏 1 𝑎 + 𝑏

𝑎 + 𝑏 2 𝑎2 + 2𝑎𝑏 + 𝑏2

𝑎 + 𝑏 3 𝑎3 + 3𝑎2𝑏 + 3𝑎𝑏2 + 𝑏3

𝑎 + 𝑏 4 𝑎4 + 4𝑎3𝑏 + 6𝑎2𝑏2 + 4𝑎𝑏3 + 𝑏4

𝑎 + 𝑏 5 𝑎5 + 5𝑎4𝑏 + 10𝑎3𝑏2 + 10𝑎2𝑏3 + 5𝑎𝑏4 + 𝑏5

𝑎 + 𝑏 6 𝑎6 + 6𝑎5𝑏 + 15𝑎4𝑏2 + 20𝑎3𝑏3 + 15𝑎2𝑏4 + 6𝑎𝑏5 + 𝑏6

In each case we observe the

following:1. There are always 𝑛 + 1 term in the expansion.

2. The exponents on 𝑎 start with 𝑛 and decrease to 0.

3. The exponents on 𝑏 start with 0 and increase to 𝑛.

4. The sum of the exponents in each term is always 𝑛.

5. If 𝑎 and 𝑏 are both positive, all terms are positive.

6. If 𝑎 is positive and 𝑏 is negative, the terms havealternating signs; those with odd powers of 𝑏 arenegative.

7. If 𝑎 is negative and 𝑏 is positive, the terms havealternating signs; those with odd powers of 𝑎 arenegative.

8. If 𝑎 and 𝑏 are both negative, all terms are positive if𝑛 is even and negative if 𝑛 is odd.

PASCAL’S TRIANGLE To discover the pattern of the numerical coefficients of

each term, we write the coefficients in the same

arrangement as in the preceding expansions.

Row 0 1

Row 1 1 1

Row 2 1 2 1

Row 3 1 3 3 1

Row 4 1 4 6 4 1

Row 5 1 5 10 10 5 1

Row 6 1 6 15 20 15 6 1

This triangular array forms what is known as

Pascal’s triangle. The row numbercorresponds to the exponent 𝑛 in the

expansion of 𝑎 + 𝑏 𝑛. The numbers in any row,

other than the first and last which are always

1, can be determined by adding the two

numbers immediately above and to the left

and right of it. Pascal’s triangle gives us one

way to determine the coefficients in the

expansion of given binomial.

Sample Problems1. Expand 2𝑥 + 𝑦 4.

In this example, 𝑛 = 4, 𝑎 = 2𝑥 and 𝑏 = 𝑦. Row 4 ofPascal’s triangle has the following coefficients

1 4 6 4 1

In 𝑎 + 𝑏 4 corresponds to expansion 1𝑎4 + 4𝑎3𝑏 +6𝑎2𝑏2 + 4𝑎𝑏3 + 1𝑏4. We obtain,

𝑎4 + 4𝑎3𝑏 + 6𝑎2𝑏2 + 4𝑎𝑏3 + 𝑏4

2𝑥 4 + 4 2𝑥 3 𝑦 + 6 2𝑥 2 𝑦 2 + 4 2𝑥 𝑦 3 + 𝑦 4

Thus,

2𝑥 + 𝑦 4 = 16𝑥4 + 32𝑥3𝑦 + 24𝑥2𝑦2 + 8𝑥𝑦3 + 𝑦4

2. Expand 𝑧 − 3 5.

In this example, 𝑛 = 5, 𝑎 = 𝑧 and 𝑏 = −3. Since we areexpanding 𝑧 + −3 5, the resulting series alternatesince 𝑏 is negative. From Row 5 of Pascal’s triangle,the coefficients are as follows: 1 5 10 10 5 1and

𝑎 + 𝑏 5 = 1𝑎5 + 5𝑎4𝑏 + 10𝑎3𝑏2 + 10𝑎2𝑏3 + 5𝑎𝑏4 + 1𝑏5

𝑧 − 3 5

= 𝑧 5 + 5 𝑧 4 −3 + 10 𝑧 3 −3 2 + 10 𝑧 2 −3 3 + 5 𝑧 −3 4

+ 𝑧 5

𝑧 − 3 5 = 𝑧5 − 15𝑧4 + 90𝑧3 − 270𝑧2 + 405𝑧4 − 243

3. Expand 𝑎2 − 2𝑏 6.

In this example, 𝑛 = 6, 𝑎 = 𝑎2 and 𝑏 = −2𝑏, form Row6 of Pascal’s triangle, the coefficients are thefollowing:

1 6 15 20 15 6 1

And𝑎 + 𝑏 6=1𝑎6 + 6𝑎5𝑏 + 15𝑎4𝑏2 + 20𝑎3𝑏3 + 15𝑎2𝑏4 + 6𝑎𝑏5 + 1𝑏6

𝑎2 − 2𝑏 6= 𝑎2 6 + 6 𝑎2 5 −2𝑏 + 15 𝑎2 4 −2𝑏 2 +20 𝑎2 3 −2𝑏 3 + 15 𝑎2 2 −2𝑏 4 + 6 𝑎2 −2𝑏 5 +−2𝑏 6

𝑎2 − 2𝑏 6 = 𝑎12 − 12𝑎10𝑏 + 60𝑎8𝑏2 − 160𝑎6𝑏3 + 240𝑎4𝑏4 − 192𝑎2𝑏5 + 64𝑏6

Break a leg!

1. Expand 𝑥−2 − 3 5.

2. Expand 𝑥2 + 𝑦 6.

THANK YOU VERY MUCH!!!PROF. DENMAR ESTRADA MARASIGAN