Other Bracket expansions Slideshow 12, Mathematics Mr Richard Sasaki

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

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.

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

Binomial Expansion The triangle refers to the coefficients of each term. Let’s expand 𝑥+𝑦 4 . 𝑥+𝑦 4 = 𝑥 4 + 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 3 +12 𝑥 2 𝑦+6𝑥 𝑦 2 + 𝑦 3

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 2 +10 𝑎 2 𝑏 3 +5𝑎 𝑏 4 + 𝑏 5 8 𝑥 3 +24 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 4 +32 𝑥 3 𝑦+24 𝑥 2 𝑦 2 +8𝑥 𝑦 3 + 𝑦 4 27 𝑥 3 +54 𝑥 2 𝑦+36𝑥 𝑦 2 +8 𝑦 3 𝑥 10 −5 𝑥 8 𝑦 2 +10 𝑥 6 𝑦 4 −10 𝑥 4 𝑦 6 +5 𝑥 2 𝑦 8 − 𝑦 10 8 𝑥 6 +12 𝑥 4 𝑦+6 𝑥 2 𝑦 2 + 𝑦 3 81 𝑥 4 −216 𝑥 3 𝑦+216 𝑥 2 𝑦 2 −96𝑥 𝑦 3 +16 𝑦 4

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+ 3 2 2 . 𝑥+ 3 2 2 = 𝑥 2 +2∙𝑥∙ 3 2 + 3 2 2 = 𝑥 2 +3𝑥+ 9 4 Note: Here we use the principle 𝑥+𝑦 2 = . 𝑥 2 +2𝑥𝑦+ 𝑦 2

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 12 + 9𝑎𝑥 12 + 6 𝑎 2 12 = 𝑥 2 + 17𝑎𝑥 12 + 𝑎 2 2 Note: Here we use the principle 𝑥+𝑎 𝑥+𝑏 = . 𝑥 2 +𝑎𝑥+𝑏𝑥+𝑎𝑏

𝑥 2 +𝑥+ 1 4 𝑥 2 − 𝑥 2 + 1 16 𝑥 2 +𝑎𝑥+ 𝑎 2 4 𝑥 2 + 3𝑥 2 + 9 16 𝑥 2 − 4𝑥 3 + 4 9 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 + 4 9𝑎 2 𝑥 2 − 8𝑥 5𝑎 + 16 25𝑎 2 𝑥 2 + 𝑥 2 + 1 18 𝑥 2 + 5𝑎𝑥 6 + 𝑎 2 6 𝑥 2 + 𝑥 9 − 2 27 𝑥 2 + 17𝑥 12 + 1 2 𝑥 2 − 1 𝑎 2 𝑥 2 + 13𝑎𝑥 6 + 𝑎 2 𝑥 2 + 𝑎𝑥 2 + 2𝑥 𝑎 +1 𝑥 2 − 23𝑥 10 + 6 5 𝑥 2 − 𝑎𝑥 28 − 𝑎 2 14 𝑥 2 + 𝑥 6𝑎 − 1 6 𝑎 2

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

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 2 + 10𝑎𝑥 5 − 6𝑎𝑥 5 − 2 𝑎 2 5 = 6𝑥 2 + 4𝑎𝑥 5 − 2 𝑎 2 5

𝑥 2 +𝑥+0.25 𝑥 2 −0.2𝑥+0.01 𝑥 2 +0.6𝑥+0.09 𝑥 2 −5𝑥+6.25 𝑥 2 +2.8𝑎𝑥+1.96 𝑎 2 𝑥 2 −0.09 𝑥 2 +0.6𝑥+0.08 𝑥 2 −2.25 𝑥 2 +0.9𝑎𝑥−4.42 𝑎 2 𝑥 2 +3.8𝑎𝑥−4.9𝑥−18.62𝑎 4𝑥 2 −2𝑥+0.25 9𝑥 2 +1.2𝑥+0.04 4𝑥 2 + 8𝑥 3 + 4 9 4𝑥 2 −2𝑎𝑥+ 𝑎 2 4 9𝑥 2 − 1 4 6𝑥 2 + 𝑎𝑥 3 − 𝑎 2 9 2𝑥 2 −0.1𝑥−0.03 8𝑥 2 +6.6𝑥+0.45 15𝑥 2 − 5𝑎𝑥 14 − 5 𝑎 2 14 6𝑥 2 − 23𝑥 28 − 15 28