Other bracket expansions

Slideshow 12, MathematicsMr Sasaki, Room 307

Objectives• Review and understand Pascal’s

triangle• Expand brackets with fractions• Expand brackets with decimals

Polynomial FormsAs you know, polynomials are in the form…

𝑎𝑥𝑚+𝑏 𝑦𝑛+…+𝑐 𝑧𝑝

And they have a finite number of terms (not infinite).Smaller polynomials have special names.𝑎𝑥𝑛

Monomial𝑎𝑥𝑚+𝑏 𝑦𝑛

Binomial𝑎𝑥𝑚+𝑏 𝑦𝑛+𝑐 𝑧𝑝

TrinomialThe focus of this lesson is multiplying binomials.Last lesson we saw a pattern named Pascal’s Triangle.

Binomial ExpansionUsing Pascal’s triangle, we can expand binomials multiplying one another that are identical like .Each level relates to expansions of .

(𝑥+ 𝑦 )0(𝑥+𝑦 )1(𝑥+𝑦 )2(𝑥+ 𝑦 )3(𝑥+𝑦 )4(𝑥+𝑦 )5(𝑥+𝑦 )6(𝑥+ 𝑦 )7(𝑥+𝑦 )8(𝑥+𝑦 )9(𝑥+ 𝑦 )10

The triangle can continue downwards further.Hopefully you understand the number pattern!

Binomial ExpansionThe triangle refers to the coefficients of each term.Let’s expand .

(𝑥+𝑦 )4=¿𝑥4+𝑥3 𝑦+𝑥2𝑦 2+𝑥 𝑦3+𝑦44 6 4When we write polynomials, it is best to write them with powers of decreasing.

(2 𝑥+ 𝑦 )3=¿(2 𝑥)3+¿3 3

¿8 𝑥3+3 ∙4 𝑥2 𝑦+3 ∙2 𝑥 𝑦 2+𝑦3¿8 𝑥3+12𝑥2 𝑦+6 𝑥 𝑦2+𝑦3

Here, we substituted for and for .

Answers1 𝑎5+5𝑎4𝑏+10𝑎3𝑏2+10𝑎2𝑏3+5𝑎𝑏4+𝑏58 𝑥3+24 𝑥2 𝑦+24 𝑥 𝑦2+8 𝑦3𝑥4−4 𝑥3 𝑦+6 𝑥2 𝑦 2−4 𝑥 𝑦3+𝑦 4

8 𝑥3−12𝑥2 𝑦+6 𝑥 𝑦2− 𝑦316 𝑥4+32 𝑥3 𝑦+24 𝑥2 𝑦2+8 𝑥 𝑦3+𝑦 4

27 𝑥3+54 𝑥2 𝑦+36 𝑥 𝑦2+8 𝑦 3𝑥10−5 𝑥8 𝑦2+10 𝑥6 𝑦4−10 𝑥4 𝑦6+5𝑥2 𝑦8− 𝑦108 𝑥6+12𝑥4 𝑦+6 𝑥2 𝑦2+𝑦 3

81 𝑥4−216 𝑥3 𝑦+216 𝑥2 𝑦2−96 𝑥 𝑦 3+16 𝑦 4

Brackets with FractionsSometimes, we also need to multiply binomials with fractions. The process is the same, just we need to think about fractions!ExampleExpand

(𝑥+32 )

2

=¿𝑥2+2∙ 𝑥 ∙ 32+( 32 )

2

¿ 𝑥2+3 𝑥+94

Note: Here we use the principle .𝑥2+2𝑥𝑦+ 𝑦2

Answers𝑥2+𝑥+

14 𝑥2− 𝑥2 +

116

𝑥2+ 3 𝑥2 +916

𝑥2+𝑎𝑥+𝑎24

𝑥2− 4 𝑥3 +49

𝑥2− 𝑎𝑥2

+𝑎216

𝑥2− 2𝑥𝑎 +1𝑎2

𝑥2− 6 𝑥𝑎 +9𝑎2

𝑥2− 4 𝑥3𝑎 +49𝑎2

𝑥2− 8 𝑥5 𝑎+1625𝑎2

Other Brackets with FractionsObviously brackets that aren’t squared work as you would expect.ExampleExpand .

(𝑥+ 2𝑎3 )(𝑥+ 3𝑎

4 )=¿𝑥2+ 2𝑎𝑥3 +3𝑎𝑥4 +

2𝑎3 ∙

3𝑎4

¿ 𝑥2+ 8𝑎𝑥12

+9𝑎𝑥12

+6 𝑎212¿ 𝑥2+ 17 𝑎𝑥

12+𝑎22

Note: Here we use the principle . 𝑥2+𝑎𝑥+𝑏𝑥+𝑎𝑏

Answers

𝑥2+ 𝑥2 +118

𝑥2+ 5𝑎𝑥6

+𝑎26

𝑥2+ 𝑥9 −227

𝑥2+ 17 𝑥12 +12

𝑥2+ 13𝑎𝑥6 +𝑎2 𝑥2− 1𝑎2

𝑥2+𝑎𝑥2 +2𝑥𝑎 +1 𝑥2− 23 𝑥10 +

65

𝑥2− 𝑎𝑥28− 𝑎

2

14𝑥2+ 𝑥

6 𝑎−16 𝑎2

Brackets with DecimalsMultiplying decimals isn’t hard!ExampleExpand .(𝑥+0.4 ) (𝑥−0.6 )=¿

¿ 𝑥2+0.4 𝑥−0.6 𝑥−(0.4 ∙0.6 )

¿ 𝑥2−0.2𝑥−0.24

Answers

𝑥2+𝑥+0.25 𝑥2−0.2𝑥+0.01

𝑥2+0.6 𝑥+0.09 𝑥2−5 𝑥+6.25

𝑥2+2.8𝑎𝑥+1.96 𝑎2𝑥2−0.09𝑥2+0.6 𝑥+0.08 𝑥2−2.25

𝑥2+0.9𝑎𝑥−4.42𝑎2𝑥2+3.8𝑎𝑥−4.9𝑥−18.62𝑎

Dealing with coefficientscoefficients other than 1 may make the calculations messier. ExampleExpand .

(2 𝑥+ 2𝑎3 )(3 𝑥− 3𝑎5 )=¿6 𝑥2+2𝑎3 ∙3 𝑥−

3𝑎5 ∙2𝑥−

2𝑎3 ∙

3𝑎5

¿6 𝑥2+2𝑎𝑥− 6𝑎𝑥5− 6𝑎

2

15¿6 𝑥2+10 𝑎𝑥

5− 6 𝑎𝑥

5− 2𝑎

2

5¿6 𝑥2+ 4 𝑎𝑥

5− 2𝑎

2

5

Answers

4 𝑥2−2 𝑥+0.25 9 𝑥2+1.2𝑥+0.044 𝑥2+ 8 𝑥3 +

49 4 𝑥2−2𝑎𝑥+

𝑎24

9 𝑥2− 14 6 𝑥2+𝑎𝑥3− 𝑎

2

9

2 𝑥2−0.1 𝑥−0.03 8 𝑥2+6.6 𝑥+0.456 𝑥2− 23 𝑥28 −

1528

15 𝑥2− 5𝑎𝑥14

− 5 𝑎2

14