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

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 Squares

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

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

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

LCM Example

Multiples

\(12 \to 12, 24, 36, 48, 60, 72, 84, 96\)

\(16 \to 16, 32, 48, 64, 80, 96\)

HCF Example

Factors

\(12 \to 2, 3, 4, 6, 12\)

\(16 \to 2, 4, 8, 16\)

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.