# Before Beginning

## Four Fours

Make numbers using four fours

• $$0=4 + 4 - 4 -4$$
• $$1=\frac{44}{44}$$
• $$2-=\frac 4 4 + \frac 4 4$$
• $$3 = \frac{4 \times 4 -4}{4} = \frac{4+4+4}{4}$$

## An Analytic Problem

Arrange 10 soldiers in 5 rows with 4 in each row.

Solution

# Chapter 4.1

Chapter name: Multiplication of Algebraic Expressions

#### Topics

• Multiplication Law
• Exponential Law
• Multiplication of Signs
• Multiplication of Monomial and Polynomial
• Exercises

## Laws of Multiplication

• Cumulative law: $$a \times b = b \times a \space , e.g., 2 \times 3 = 3 \times 2$$
• Associative law: $$(a \times b) \times c = a \times (b \times c), e.g., (2 \times 3) \times 4 = 2 \times (3 \times 4)$$
• Exponential law: $$a \times a \times a = a^3, a^2 \times a^4 = a^6$$
• $$a^m \times a^n = a^{m+n}, (a^m)^n = a^{mn}$$
• Distributive law: 2(a+b) = 2a + 2b
• Caution: $$2^3 \ne 2 \times 3 = 6$$, but $$2^3 = 2 \times 2 \times 2$$

## Sign Laws

• $$(+1) \times (+1) = +1$$
• $$(+1) \times (-1) = -1$$
• $$(-1) \times (+1) = -1$$
• $$(-1) \times (-1) = +1$$
• $$+ve \rightarrow friend$$

• $$-ve \rightarrow foe$$

• $$(+1) \times (+1) \rightarrow$$ friend of friend
• $$(+1) \times (-1) \rightarrow$$ friend of foe

## Problems

Multiply

1. $$5a^2x^2$$ with $$3ax^5y$$
2. $$2x+3y$$ with $$5xy$$
3. $$x^2+2xy+y^2$$ with $$x+y$$
4. $$x^2+xy+y^2$$ with $$x^2-xy+y^2$$
5. $$y^2-y+1$$ with $$1+y+y^2$$
6. $$2^{-3}x^5y^3$$ with $$2^3 x^{-4}y^{-1}$$
1. $$15a^3x^7y$$
1. $$xy^2$$

## Chapter 4.2: Division

Divide the first expression by the second one

1. $$30a^4x^3, -6a^2x$$
2. $$36x^4y^3+9x^5y^2, 9xy$$
3. $$6x^2+x-2, 2x-1$$
4. $$6y^2+3x^2-11xy, 3x-2y$$
5. $$x^4y^4-1, x^2y^2+1$$
6. $$a^5+11a-12, a^2-2a+3$$
1. $$-5a^2x^2$$
1. $$4x^3y^2+x^4y$$
1. $$3x+2$$
1. $$x-3y$$
1. $$x^2y^2-1$$
1. $$a^3+2a^2+a-4$$

## Div

Questions

1. $$30a^4x^3, -6a^2x$$
2.$$36x^4y^3+9x^5y^2, 9xy$$

# 4.3: Simplification

## Rule of PEMDAS/BEDMAS/BODMAS

Notation varies by countries

• $$P/B \rightarrow$$ ()
• $$E/O \rightarrow$$ Exponent/Order $$\rightarrow 2^3$$
• MD $$\rightarrow$$ Multiplication/Division
• AS $$\rightarrow$$ Addition/Subtraction
• BO $$\ne$$ Bracket OFF! O is order/exponent
• $$3 \div \frac 1 2=?$$ $$3 \div 1\div 2=?$$
• $$6 \times 3 \div3=?$$
• $$(2+3)^2 + 6 \times 3 \div3 + 4-3=?$$

## Simplify

Questions

• $$1. 6-2\{5-(8-3)+(5+2)\}$$
• $$2. 7-2[-6+3\{-5+2(4-3)\}]$$
• $$3. 3x+(4y-z)-{a-b-(2c-4a)-5a}$$
• $$4. [8b-3\{2a-3(2b+5)-5(b-3)\}]-3b$$

## Simplify (cntd.)

$$A=x^2-xy+y^2$$

$$B = x^2+xy+y^2$$

$$C = x^4+x^2y^2+y^4$$

1. Find $$A \times B$$
1. Determine $$BC \div B^2-C$$

# Formulae

MEMORIZE FEEL

• $$(a+b)^2 = a^2+2ab+b^2$$
• $$(a-b)^2 = a^2-2ab+b^2$$
• $$a^2-b^2=(a+b)(a-b)$$

# Corollary

• $$a^2+b^2= ?$$
• $$a^2+b^2= ?$$ (in another form)
• $$(a+b)^2 = ?$$ (in terms of) $$(a-b)^2$$
• $$(a-b)^2 = ?$$ (in terms of) $$(a+b)^2$$
• $$(a+b)^2+(a-b)^2 = ?$$
• $$(a+b)^2+(a-b)^2 = ?$$
• $$ab=?$$

# Geometric Interpretation

$$(a+b)^2 = a^2+2ab+b^2$$

# Relationship between (a+b)2 and (a-b)2

$\begin{eqnarray} (a+b)^2 &=& a^2+2ab+b^2 \nonumber \\ &=& a^2-2ab+b^2 + 2ab + 2ab \nonumber \\ &=& (a-b)^2 +4ab \nonumber \\ \end{eqnarray}$

• Find (a-b)2 similarly

• (2x+y)
• (5m-3p)
• ax+b+2

# Simplify

Questions

• $$1.\space (x+y)^2+2(x+y)(x-y)+(x-y)^2$$
• $$2. \space (2a+1)^2-4a(2a+1)+4a^2$$
• $$3. \space (5a+3b)^2+2(5a+3b)(4a-3b)+(4a-3b)^2$$
• $$4. \space (8x+y)^2-(16x+2y)(5x+y)+(5x+y)^2$$

# Find Values Using Formula

Questions

• $$1.\space 25x^2+36y^2-60xy$$ if $$x=-4, y=-5$$
• $$2. \space if \space (a-b)=7$$ and $$ab=3, (a+b)^2=?$$
• $$3. \space if \space x+\frac 1 x = 5$$ find $$(x^2-\frac 1 {x^2})^2$$
• $$4. \space m+\frac 1 m = 2, m^4+\frac 1 {m^4}=?$$
• $$5. \space p^2-3p+1=0, p^2+\frac 1 {p^2}=?$$

# Formula

• $$(a+b)(a-b)=a^2-b^2$$
• $$(x+a)(x+b) = x^2+x(a+b)+ab$$

# Multiply Using Formula

Questions

• $$1. \space (10+xy)(10-xy)$$
• $$2. \space (3x+2y+1)(3x-2y+1)$$
• $$3. \space (x^2-x+1(x^2+x+1)$$
• $$4. \space (y+4)(y+7)$$

# Chapter 5.3 (Factorization)

## Factorization Concept

• What is a factor? What’s another name?
• $$12 = 3 \times 4 = 3 \times 2 \times 2$$
• $$a^2+ab = a(a+b)$$
• $$a^2-b^2 = (a+b)(a-b)$$
• $$a^2+5a+6=?$$

## Resolve into Factors

Questions

• $$1. \space 2x-6x^2$$
• $$2. \space 25(a+2b)^2-36(2a-5b)^2$$
• $$3. \space 2bd-a^2-c^2+b^2+d^2+2ac$$
• $$4. \space x^2+11x+30$$
• $$5. \space 2x^2+11x+12$$
• $$6. \space 6x^2+17x+5$$

# Chapter 5.2: LCM-HCF

• Factor
• Dividend
• Quotient
• Divisor

## LCM Example

Multiples

$$12 \to 12, 24, 36, 48, 60, 72, 84, 96$$

$$16 \to 16, 32, 48, 64, 80, 96$$

• Common multiples: 48, 96
• Lowest Common multiple: 48
• $$8x^2yz^2, 10x^3yz^3$$

## HCF Example

Factors

$$12 \to 2, 3, 4, 6, 12$$

$$16 \to 2, 4, 8, 16$$

• Common factors: 2, 4
• Highest common factor (HCF): 4

## Find LCM

Questions

• $$1. 4x^2y^3z, 6xy^3z^2, 8x^3yz^3$$
• $$2. a-2, a^2-4, a^2-a-2$$
• $$3. x^3-3x^2-10x, x^3+6x^2+8x, x^4-5x^3-14x^2$$
• $$4. xy-y, x^3y-xy, x^2-2x+1$$
• $$5. x^2-8x+15, x^2-25, x^2+2x-15$$
• $$6. a^2-7a+12, a^2+a-20, a^2+2a-15$$

## Find HCF

Questions

• $$1. 4x^2y^3z, 6xy^3z^2, 8x^3yz^3$$
• $$2. a^2+ab, a^2-b^2$$
• $$3. x^3-3x^2-10x, x^3+6x^2+8x, x^4-5x^3-14x^2$$
• $$4. xy-y, x^3y-xy, x^2-2x+1$$
• $$5. x^2-8x+15, x^2-25, x^2+2x-15$$
• $$6. a^2-7a+12, a^2+a-20, a^2+2a-15$$

## Can HCF be > LCM

Consider 0, 9

Factors

• $$0 \times 1 = 0$$
• $$0 \times 2 = 0$$
• $$0 \times 3 = 0$$

$$\cdots$$

$$\cdots$$

Factors of 0: 0, 1, 2, 3, …, 9, 10,…

Factors of 9: 1, 3, 9

Multiples

We get multiples by multiplying the numbers by $$1, 2, 3, \cdots$$

• $$0 \times 1 = 0$$
• $$0 \times 2 = 0$$
• $$0 \times 3 = 0$$

Multiple of 0: 0

Multiple of 9: 0, 9, 18, 27, …(0 is a multiple of any number)

• HCF = 9
• LCM = 0

# Chapter 5.2: Algebraic Fractions

## Reduce to Lowest Form

Questions

• $$1.\frac{a^2b}{a^3b}$$
• $$2. \frac{x^2+x}{xy+y}$$
• $$3. \frac{x^2+2x-15}{x^2+9x+20}$$

## Express with Common Denominator

Questions

• $$1. \frac{a}{bc}, \frac{a}{ac}$$
• $$2. \frac{a}{a-b}, \frac{b}{a+b}$$
• $$3. \frac{3}{a^2-4}, \frac{2}{a(a+2)}$$
• $$4. \frac{2}{x^2-x-2}, \frac{3}{x^2+x-6}$$

Questions

• $$1. \frac{x}{2a} + \frac{y}{3b}$$
• $$2. \frac{3}{x^2-4x-5} + \frac{4}{x+1}$$
• $$3. \frac{3}{x+3}-\frac{2}{x+2}$$
• $$4. \frac{1}{x^2-1}+\frac 1 {x^4-1}+\frac 4 {x^8-1}$$