VII Algebra

Abdullah Al Mahmud

VII Algebra Abdullah Al Mahmud

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

$(5x)^2-2 \times 5x \times 6y + (6y)^2 = (5x-6y)^2 \to 100$

$(a-b)^2+4ab = 7^2+4 \times 3$

$(x^2+\frac 1 {x^2})^2-4 \times x^2 \times \frac 1 {x^2} \cdots \to 525$

Two ways to solve $\to or \leftarrow$

$p^2+1=3p \to \frac {p^2+1}{p}=3 \to \cdots$

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

$2x(1-3x)$

$(5x)^2-(6y)^2$

$(b+d)^2-(a-c)^2$

$x^2+5x+6x+30$

$2x^2+8x+3x+12 \to$

$6x^2+15x+2x+5 \to$

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

$24x^3y^2z^3$

$(a+1)(a+2)(a-2)$

$x^2(x+2)(x+4)(x-7)$

$xy(x+1)(x-1)^2$

$(x-3)(x^2-25)$

$(a+5)(a-3)(a-4)$

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

$2xyz$

$(a+b)$

$x(x+2)$

$(x-1)$

Can HCF be > LCM

Consider 0, 9

Factors

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

$\cdots$

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

$\frac x y$

$\frac{(x-3)}{(x+4)}$

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

$\frac{a^2}{ab}, \frac{ab}{abc}$

$\frac{a(a+b)}{(a^2-b^2)}, \frac{b(a-b)}{(a^2-b^2)}$

$\frac{3a}{a(a^2-4)}, \frac{2(a-2)}{a(a^2-4)}$

$\frac{2(x+3)}{(x-2)(x+1)(x+3)}, \frac{3(x+1)}{(x-2)(x+1)(x+3)}$

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

$\frac{3+2ay}{6ab}$

$\frac{4x-17}{(x-5)(x+1)}$

$\frac x {(x+2)(x+3)}$

$\frac{x^6+2x^4+x^2+6}{(x^8-1)}$

VII Algebra

Abdullah Al Mahmud

Geometric Interpretation

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

Find Squares

Simplify

Find Values Using Formula

Chapter 5.2

Formula

Multiply Using Formula

Chapter 5.3 (Factorization)

Factorization Concept

Resolve into Factors

Chapter 5.2: LCM-HCF

LCM-HCF Concepts

LCM Example

HCF Example

Find LCM

Find HCF

Can HCF be > LCM

Chapter 5.2: Algebraic Fractions

Reduce to Lowest Form

Express with Common Denominator

Addistion-Subtraction of Algebraic Fractions

Problems to Think

Camel Problem

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