The document discusses binomial expansions and Pascal's triangle. It explains that when a binomial of the form a + b is raised to a power n, the resulting polynomial expansion will have n + 1 terms with the exponents of a decreasing from n to 0 and exponents of b increasing from 0 to n. Pascal's triangle provides the coefficients of the terms, with each row corresponding to the exponents in the expansion of a + b to the power of that row number. The document provides examples of expanding binomials like (2x + y)^4 and (z - 3)^5 using the patterns of exponents and Pascal's triangle.

1. BINOMIAL
EXPANSION

2. 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 + 3𝑎2 𝑏 + 3𝑎𝑏2 + 𝑏3
𝑎 + 𝑏 4
𝑎4
+ 4𝑎3
𝑏 + 6𝑎2
𝑏2
+ 4𝑎𝑏3
+ 𝑏4
𝑎 + 𝑏 5 𝑎5 + 5𝑎4 𝑏 + 10𝑎3 𝑏2 + 10𝑎2 𝑏3 + 5𝑎𝑏4 + 𝑏5
𝑎 + 𝑏 6
𝑎6
+ 6𝑎5
𝑏 + 15𝑎4
𝑏2
+ 20𝑎3
𝑏3
+ 15𝑎2
𝑏4
+ 6𝑎𝑏5
+ 𝑏6

3. 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 have
alternating signs; those with odd powers of 𝑏 are
negative.
7. If 𝑎 is negative and 𝑏 is positive, the terms have
alternating signs; those with odd powers of 𝑎 are
negative.
8. If 𝑎 and 𝑏 are both negative, all terms are positive if
𝑛 is even and negative if 𝑛 is odd.

4. 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

5.  This triangular array forms what is known as
Pascal’s triangle. The row number
corresponds 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.

6. Sample Problems
1. Expand 2𝑥 + 𝑦 4.
In this example, 𝑛 = 4, 𝑎 = 2𝑥 and 𝑏 = 𝑦. Row 4 of
Pascal’s triangle has the following coefficients
1 4 6 4 1
In 𝑎 + 𝑏 4 corresponds to expansion 1𝑎4 + 4𝑎3 𝑏 +
6𝑎2 𝑏2 + 4𝑎𝑏3 + 1𝑏4. We obtain,
𝑎4 + 4𝑎3 𝑏 + 6𝑎2 𝑏2 + 4𝑎𝑏3 + 𝑏4
2𝑥 4 + 4 2𝑥 3 𝑦 + 6 2𝑥 2 𝑦 2 + 4 2𝑥 𝑦 3 + 𝑦 4
Thus,
2𝑥 + 𝑦 4
= 16𝑥4
+ 32𝑥3
𝑦 + 24𝑥2
𝑦2
+ 8𝑥𝑦3
+ 𝑦4

7. 2. Expand 𝑧 − 3 5.
In this example, 𝑛 = 5, 𝑎 = 𝑧 and 𝑏 = −3. Since we are
expanding 𝑧 + −3 5, the resulting series alternate
since 𝑏 is negative. From Row 5 of Pascal’s triangle,
the coefficients are as follows: 1 5 10 10 5 1
and
𝑎 + 𝑏 5
= 1𝑎5
+ 5𝑎4
𝑏 + 10𝑎3
𝑏2
+ 10𝑎2
𝑏3
+ 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

8. 3. Expand 𝑎2 − 2𝑏 6.
In this example, 𝑛 = 6, 𝑎 = 𝑎2 and 𝑏 = −2𝑏, form Row
6 of Pascal’s triangle, the coefficients are the
following:
1 6 15 20 15 6 1
And
𝑎 + 𝑏 6
=1𝑎6
+ 6𝑎5
𝑏 + 15𝑎4
𝑏2
+ 20𝑎3
𝑏3
+ 15𝑎2
𝑏4
+ 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
− 12𝑎10
𝑏 + 60𝑎8
𝑏2
− 160𝑎6
𝑏3
+ 240𝑎4
𝑏4
− 192𝑎2
𝑏5
+ 64𝑏6

9. Break a leg!
1. Expand 𝑥−2 − 3 5.
2. Expand 𝑥2
+ 𝑦 6
.

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