Bracket Expansion

Slideshow 11, Mathematics

Mr Sasaki, Room 307

Objectives• To recall how to multiply a polynomial

by a number or monomial• Recall how to multiply two polynomials

Vocabulary ReviewWe need to review some vocabulary regarding polynomials.

𝑎𝑥𝑏(𝑚 𝑦𝑛+𝑝 𝑧𝑞)Monomial(An expression with one term)

Polynomial

TermsWe need to review some vocabulary regarding polynomials.

Multiplying Polynomials by a MonomialWhen we multiply a monomial by a polynomial, we multiply the monomial by each term.ExampleExpand and simplify 3 𝑥2 (2 𝑥−4 )=¿6 𝑥3−12𝑥2

Wow, that was hard, right?

Note: Don’t forget laws of indices…𝑥𝑎×𝑥𝑏=¿𝑥𝑎+𝑏𝑥𝑎÷ 𝑥𝑏=¿𝑥𝑎−𝑏

Answers – Top Part3 𝑥+6 2 𝑥−2 𝑦 8 𝑥+12 𝑦−𝑥+3 16 𝑥2+24−6 𝑦3+3 𝑦2

4 𝑥3 8 𝑥2+6 𝑥𝑦 18 𝑥−4 𝑥2

3 𝑥2+410 𝑥+4 𝑦2

−6 𝑥2 𝑦 12𝑥4−6 𝑥33 𝑥3+6 𝑥2−𝑥3−𝑥22 −3 𝑥

32 −3 𝑦−12

Answers – Bottom Part12𝑥3+8𝑥2−6 𝑥𝑦−10 𝑦2 28 𝑥3+21𝑥2

4 𝑥+7 2 𝑥2+4 𝑦 2 𝑥+4 𝑦8 𝑥3+16 𝑥2𝑦−20𝑥2𝑧 210 𝑥2𝑦−20𝑥 𝑦25 𝑥 𝑦 2+2 𝑦4 𝑥𝑦−12𝑥412𝑥2+3 𝑥 12𝑥 𝑦2−6 𝑥2

14 𝑥3 𝑦3 𝑧+21 𝑥2𝑦 230 𝑥2 𝑦−20𝑥324 𝑥 𝑦2−6 𝑥3 𝑦3

2 𝑥2+4 𝑥𝑥5

2+𝑥4

(6 𝑥¿¿5+12𝑥4)𝑦 ¿

Multiplying PolynomialsAs you know, when you multiply polynomials together, you must consider each possible product from the terms.Write down the expansion of .

(𝑎+𝑏+𝑐 ) (𝑥+ 𝑦+𝑧 )=¿𝑎𝑥+𝑎𝑦+𝑎𝑧+¿𝑏𝑥+𝑏𝑦 +𝑏𝑧+¿𝑐𝑥+𝑐𝑦+𝑐𝑧

Try to be systematic so that you don’t miss anything!

Multiplying PolynomialsExamplesExpand.

15 𝑥3+10 𝑥2−6 𝑥𝑦−4 𝑦Expand.+2 𝑥4−2 𝑥5 −𝑥−2+𝑥−1Expand.

It may be easier to multiply the monomial into the expression first.

(6 𝑥2+3 𝑥𝑦 ) (2 𝑦−𝑧 )=¿12𝑥2 𝑦−6 𝑥2𝑧+6 𝑥 𝑦2−3 𝑥𝑦𝑧

Answers – Top Part

𝑥2+5 𝑥+4 𝑥2−𝑥−6 𝑥2+6 𝑥+9

𝑥2−16 𝑥+64𝑥2−9 2 𝑥2−7 𝑥−4

𝑥3−3𝑥2+4 𝑥−123 𝑥2−10 𝑥+3𝑥𝑦−2𝑥−3 𝑦+6

3 𝑥2+4 𝑥−43 𝑥2−6𝑥−24

Answers – Bottom Part2 𝑥2+15 𝑥+7 9 𝑥2−6𝑥+18 𝑥2−14 𝑥+6 −2 𝑥2+7𝑥+15𝑥𝑦+4 𝑥− 𝑦2− 𝑦+12 2 𝑥2+5 𝑥𝑦−4 𝑥+2 𝑦2−8 𝑦8 𝑥2+2𝑥𝑦−4 𝑥𝑧− 𝑦2+4 𝑦𝑧−4 𝑧 248𝑥2 𝑦+24 𝑥𝑦+3 𝑦8 𝑥𝑦𝑧−20 𝑥𝑦+16 𝑦𝑧−40 𝑦2 𝑥3 𝑦 2−2𝑥3+4 𝑥2 𝑦2−4 𝑥2

18 𝑥3−6𝑥2 𝑦+42𝑥2−12𝑥 𝑦2+48 𝑥𝑦−36 𝑥27 𝑥4−12 𝑥2−𝑥2+𝑥+𝑥−1−1 4 𝑥3+16𝑥2+16 𝑥−1+4

6 𝑥3+18 𝑥+6𝑥−1