other bracket expansions

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Other bracket expansions Slideshow 12, Mathematics Mr Sasaki, Room 307

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Other bracket expansions. Slideshow 12 Mathematics Mr Sasaki Room 307. Objectives. Review last lesson’s pattern Expand brackets with fractions Expand brackets with decimals. Binomial Expansion. When we have a polynomial with two terms, we call it a binomial . (like or ). - PowerPoint PPT Presentation

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Page 1: Other bracket expansions

Other bracket expansions

Slideshow 12, MathematicsMr Sasaki, Room 307

Page 2: Other bracket expansions

Objectives• Review and understand Pascal’s

triangle• Expand brackets with fractions• Expand brackets with decimals

Page 3: Other bracket expansions

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.

Page 4: Other bracket expansions

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!

Page 5: Other bracket expansions

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 .

Page 6: Other bracket expansions

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

Page 7: Other bracket expansions

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

Page 8: Other bracket expansions

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

Page 9: Other bracket expansions

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

Page 10: Other bracket expansions

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

Page 11: Other bracket expansions

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

Page 12: Other bracket expansions

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

Page 13: Other bracket expansions

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

Page 14: Other bracket expansions

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