other bracket expansions
DESCRIPTION
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 PresentationTRANSCRIPT
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