VII Algebra

Abdullah Al Mahmud

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.\(36x^4y^3+9x^5y^2, 9xy\)

Answers

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

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

Answers

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 \(BC \div B^2-C\)

Answers

Chapter 5.1: Algebraic Formulae & Applications

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

Find Squares

  • (2x+y)
  • (5m-3p)
  • ax+b+2

Simplify

Questions

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

Answers

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 \((x^2-\frac 1 {x^2})^2\)
  • \(4. \space m+\frac 1 m = 2, m^4+\frac 1 {m^4}=?\)
  • \(5. \space p^2-3p+1=0, p^2+\frac 1 {p^2}=?\)

Answers

Chapter 5.2

Formula

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

Multiply Using Formula

Questions

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

Answers

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

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

Answers

Chapter 5.2: LCM-HCF

LCM-HCF Concepts

  • 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

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

Answers

Find HCF

Questions

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

Answers

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

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

Answers

Express with Common Denominator

Questions

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

Answers

Addistion-Subtraction of Algebraic Fractions

Questions

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

Answers

Problems to Think

Camel Problem

Distribute 35 camels among 3 brothers so the eldest brother gets half, second brother gets one-third, and the youngest brother gets one-ninth.