1. Other Bracket expansions
Slideshow 12, Mathematics
Mr Richard Sasaki

2. Objectives Review and understand Pascal’s triangle
Expand brackets with fractions
Expand brackets with decimals

3. Polynomial Forms As you know, polynomials are in the form…
𝑎 𝑥 𝑚 +𝑏 𝑦 𝑛 +…+𝑐 𝑧 𝑝
And they have a finite number of terms (not infinite).
Smaller polynomials have special names.
𝑎 𝑥 𝑛
𝑎 𝑥 𝑚 +𝑏 𝑦 𝑛
𝑎 𝑥 𝑚 +𝑏 𝑦 𝑛 +𝑐 𝑧 𝑝
Monomial
Binomial
Trinomial
The focus of this lesson is multiplying binomials.
Last lesson we saw a pattern named Pascal’s Triangle.

4. Binomial Expansion
Using Pascal’s triangle, we can expand binomials multiplying one another that are identical like 𝑥+𝑦.
Each level relates to expansions of 𝑥+𝑦.
𝑥+𝑦 0
The triangle can continue downwards further.
𝑥+𝑦 1
𝑥+𝑦 2
𝑥+𝑦 3
𝑥+𝑦 4
𝑥+𝑦 5
Hopefully you understand the number pattern!
𝑥+𝑦 6
𝑥+𝑦 7
𝑥+𝑦 8
𝑥+𝑦 9
𝑥+𝑦 10

5. Binomial Expansion
The triangle refers to the coefficients of each term.
Let’s expand 𝑥+𝑦 4 .
𝑥+𝑦 4 =
𝑥 𝑥 3 𝑦+ 𝑥 2 𝑦 2 + 𝑥 𝑦 3 + 𝑦 4 4 6 4
When we write polynomials, it is best to write them with powers of 𝑥 decreasing.
2𝑥+𝑦 3 =
(2𝑥) 3 + (2 𝑥) 2 𝑦+ (2𝑥) 𝑦 2 + 𝑦 3 3 3
Here, we substituted 𝑥 for 2𝑥 and 𝑦 for 𝑦.
=8 𝑥 3 +3∙4 𝑥 2 𝑦+3∙2𝑥 𝑦 2 + 𝑦 3
=8 𝑥 𝑥 2 𝑦+6𝑥 𝑦 2 + 𝑦 3

6. Answers 1 𝑎 5 +5 𝑎 4 𝑏+10 𝑎 3 𝑏 2 +10 𝑎 2 𝑏 3 +5𝑎 𝑏 4 + 𝑏 5
𝑎 5 +5 𝑎 4 𝑏+10 𝑎 3 𝑏 𝑎 2 𝑏 3 +5𝑎 𝑏 4 + 𝑏 5
8 𝑥 𝑥 2 𝑦+24𝑥 𝑦 2 +8 𝑦 3
𝑥 4 −4 𝑥 3 𝑦+6 𝑥 2 𝑦 2 −4𝑥 𝑦 3 + 𝑦 4
8 𝑥 3 −12 𝑥 2 𝑦+6𝑥 𝑦 2 − 𝑦 3
16 𝑥 𝑥 3 𝑦+24 𝑥 2 𝑦 2 +8𝑥 𝑦 3 + 𝑦 4
27 𝑥 𝑥 2 𝑦+36𝑥 𝑦 2 +8 𝑦 3
𝑥 10 −5 𝑥 8 𝑦 𝑥 6 𝑦 4 −10 𝑥 4 𝑦 6 +5 𝑥 2 𝑦 8 − 𝑦 10
8 𝑥 𝑥 4 𝑦+6 𝑥 2 𝑦 2 + 𝑦 3
81 𝑥 4 −216 𝑥 3 𝑦+216 𝑥 2 𝑦 2 −96𝑥 𝑦 𝑦 4

7. Brackets with Fractions
Sometimes, we also need to multiply binomials with fractions. The process is the same, just we need to think about fractions!
Example
Expand 𝑥
𝑥 =
𝑥 2 +2∙𝑥∙
= 𝑥 2 +3𝑥+ 9 4
Note: Here we use the principle 𝑥+𝑦 2 =
𝑥 2 +2𝑥𝑦+ 𝑦 2

8. Other Brackets with Fractions
Obviously brackets that aren’t squared work as you would expect.
Example
Expand 𝑥+ 2𝑎 3 𝑥+ 3𝑎 4 .
𝑥+ 2𝑎 3 𝑥+ 3𝑎 4 =
𝑥 2 + 2𝑎𝑥 3 + 3𝑎𝑥 4 + 2𝑎 3 ∙ 3𝑎 4
= 𝑥 2 + 8𝑎𝑥 𝑎𝑥 𝑎 2 12
= 𝑥 𝑎𝑥 𝑎 2 2
Note: Here we use the principle 𝑥+𝑎 𝑥+𝑏 =
𝑥 2 +𝑎𝑥+𝑏𝑥+𝑎𝑏

9. 𝑥 2 +𝑥+ 1 4
𝑥 2 − 𝑥
𝑥 2 +𝑎𝑥+ 𝑎 2 4
𝑥 2 + 3𝑥
𝑥 2 − 4𝑥
𝑥 2 − 𝑎𝑥 2 + 𝑎 2 16
𝑥 2 − 2𝑥 𝑎 + 1 𝑎 2
𝑥 2 − 6𝑥 𝑎 + 9 𝑎 2
𝑥 2 − 4𝑥 3𝑎 𝑎 2
𝑥 2 − 8𝑥 5𝑎 𝑎 2
𝑥 2 + 𝑥
𝑥 2 + 5𝑎𝑥 6 + 𝑎 2 6
𝑥 2 + 𝑥 9 − 2 27
𝑥 𝑥
𝑥 2 − 1 𝑎 2
𝑥 𝑎𝑥 6 + 𝑎 2
𝑥 2 + 𝑎𝑥 2 + 2𝑥 𝑎 +1
𝑥 2 − 23𝑥
𝑥 2 − 𝑎𝑥 28 − 𝑎 2 14
𝑥 2 + 𝑥 6𝑎 − 1 6 𝑎 2

10. Brackets with Decimals
Multiplying decimals isn’t hard!
Example
Expand 𝑥+0.4 𝑥−0.6 .
𝑥+0.4 𝑥−0.6 =
= 𝑥 𝑥−0.6𝑥−(0.4∙0.6)
= 𝑥 2 −0.2𝑥−0.24

11. Dealing with 𝑥−coefficients
𝑥−coefficients other than 1 may make the calculations messier.
Example
Expand 2𝑥+ 2𝑎 3 3𝑥− 3𝑎 5 .
2𝑥+ 2𝑎 3 3𝑥− 3𝑎 5 =
6 𝑥 2 + 2𝑎 3 ∙3𝑥− 3𝑎 5 ∙2𝑥− 2𝑎 3 ∙ 3𝑎 5
= 6𝑥 2 +2𝑎𝑥− 6𝑎𝑥 5 − 6 𝑎 2 15
= 6𝑥 𝑎𝑥 5 − 6𝑎𝑥 5 − 2 𝑎 2 5
= 6𝑥 2 + 4𝑎𝑥 5 − 2 𝑎 2 5

12. 𝑥 2 +𝑥+0.25
𝑥 2 −0.2𝑥+0.01
𝑥 𝑥+0.09
𝑥 2 −5𝑥+6.25
𝑥 𝑎𝑥+1.96 𝑎 2
𝑥 2 −0.09
𝑥 𝑥+0.08
𝑥 2 −2.25
𝑥 𝑎𝑥−4.42 𝑎 2
𝑥 𝑎𝑥−4.9𝑥−18.62𝑎
4𝑥 2 −2𝑥+0.25
9𝑥 𝑥+0.04
4𝑥 2 + 8𝑥
4𝑥 2 −2𝑎𝑥+ 𝑎 2 4
9𝑥 2 − 1 4
6𝑥 2 + 𝑎𝑥 3 − 𝑎 2 9
2𝑥 2 −0.1𝑥−0.03
8𝑥 𝑥+0.45
15𝑥 2 − 5𝑎𝑥 14 − 5 𝑎 2 14
6𝑥 2 − 23𝑥 28 − 15 28