Rules of Positive and Negative Signs
$\displaystyle \small (+)\times (+)=+$
$\displaystyle \small (+)\times (-)=-$
$\displaystyle \small (-)\times (+)=-$
$\displaystyle \small (-)\times (-)=+$
$\displaystyle \small \frac{(+)}{(+)}=+$
$\displaystyle \small \frac{(+)}{(-)}=-$
$\displaystyle \small \frac{(-)}{(+)}=-$
$\displaystyle \small \frac{(-)}{(-)}=+$
Laws of Indices
1. $\displaystyle \small a^{n}=x$
where,
a = base
n = power
x = value of $\displaystyle \small a^{n}$
2. $\displaystyle \small a^{n}.a^{m}=a^{n+m}$
3. $\displaystyle \small a^{n}\div a^{m}=a^{n-m}$
4. $\displaystyle \small (a^{n})^{m}=a^{n.m}$
5. $\displaystyle \small (a.b)^{n}=a^{n}.b^{n}$
6. $\displaystyle \small a^{-n}=\frac{1}{a^{n}}$
7. $\displaystyle \small a^{0}=1$
8. $\displaystyle \small a^{1}=a$
9. $\displaystyle \small \frac{0}{a}=0$
10. $\displaystyle \small \frac{a}{0}=\infty$
Addition and Subtraction of Algebraic Numbers
A term in an algebraic expression involves letters and/or numbers multiplied together.
Ex: 5x [5 is coefficient and x is variable]
-8xy [-8 is coefficient and xy are varibales]
$\displaystyle \small 7x^{2}-5y$ [7 is coefficient of $\displaystyle \small x^{2}$ and -5 is coefficient of y]
We can only add or subtract like terms (like terms have same variables).
First, group the like terms and add or subtract their coefficients.
(i) Simplify 13x + 7y − 2x + 6a
group like terms,
⇒ (13x-2x)+7y+6a
⇒ 11x+7y+6a
(ii) Add (7x+2) and (2x+3)
⇒ 7x+2+2x+3
group like terms,
⇒ (7x+2x)+(2+3)
⇒ 9x+5
(iii) Add $\displaystyle \small (2x^{3}+x^{2}+x)$ and $\displaystyle \small (4x^{3}+2x^{2})$
⇒ $\displaystyle \small 2x^{3}+x^{2}+x+4x^{3}+2x^{2}$
group like terms,
⇒ $\displaystyle \small (2x^{3}+4x^{3})+(x^{2}+2x^{2})+x$
⇒ $\displaystyle \small 6x^{3}+3x^{2}+x$
(iv) Subtract $\displaystyle \small (2x^{2}-4x-1)$ from $\displaystyle \small (6x^{2}-3x+5)$
⇒ $\displaystyle \small 6x^{2}-3x+5-(2x^{2}-4x-1)$
⇒ $\displaystyle \small 6x^{2}-3x+5-2x^{2}+4x+1$
⇒ $\displaystyle \small (6x^{2}-2x^{2})+(-3x+4x)+(5+1)$
⇒ $\displaystyle \small 4x^{2}+x+6$
Multiplication and Division of Algebraic Numbers
Multiplication: Multiply the coefficients and multiply the variables using laws of indices
(i) $\displaystyle \small (6xy)\times (-3x^{2}y^{3})$
⇒ $\displaystyle \small \left \{ 6\times (-3) \right \}$ $\displaystyle \small \times \left \{ xy.x^{2}y^{3} \right \}$
⇒ $\displaystyle \small \left \{ -18 \right \}$ $\displaystyle \small \times \left \{ x^{3}y^{4} \right \}$
⇒ $\displaystyle \small -18x^{3}y^{4}$
(ii) $\displaystyle \small (5x^{2}-2y^{3})(2x^{3}+5y^{4})$
⇒ $\displaystyle \small 5x^{2}(2x^{3}+5y^{4})$ $\displaystyle \small -2y^{3}(2x^{3}+5y^{4})$
⇒ $\displaystyle \small 5x^{2}.2x^{3}+5x^{2}.5y^{4}$ $\displaystyle \small -2y^{3}.2x^{3}-2y^{3}.5y^{4}$
⇒ $\displaystyle \small 10x^{5}+25x^{2}y^{4}-4x^{3}y^{3}-10y^{7}$
Division:
Algebraic Formulae
1. $\displaystyle \small (a+b)^{2}=a^{2}+b^{2}+2ab$
2. $\displaystyle \small (a-b)^{2}=a^{2}+b^{2}-2ab$
3. $\displaystyle \small (a+b)^{2}=(a-b)^{2}+4ab$
4. $\displaystyle \small (a-b)^{2}=(a+b)^{2}-4ab$
5. $\displaystyle \small (a+b)(a-b)=a^{2}-b^{2}$
6. $\displaystyle \small (a+b)^{3}=a^{3}+b^{3}+3ab(a+b)$
7. $\displaystyle \small (a-b)^{3}=a^{3}-b^{3}-3ab(a-b)$
8. $\displaystyle \small a^{3}+b^{3}=(a+b)(a^{2}+b^{2}-ab)$
9. $\displaystyle \small a^{3}-b^{3}=(a-b)(a^{2}+b^{2}+ab)$
10. $\displaystyle \small (a+b+c)^{2}=a^{2}+b^{2}$ $\displaystyle \small +c^{2}+2ab+2bc+2ca$
Factors of Trinomial
Ex: $\displaystyle \small x^{8}-\frac{1}{x^{8}}$
⇒ $\displaystyle \small (x^{4})^{2}-\left (\frac{1}{x^{4}} \right )^{2}$
[ $\displaystyle \small a^{2}-b^{2}=(a+b)(a-b)$]
⇒ $\displaystyle \small \left ( x^{4}-\frac{1}{x^{4}} \right )\left ( x^{4}+\frac{1}{x^{4}} \right )$
⇒ $\displaystyle \small \left [(x^{2})^{2}-\left (\frac{1}{x^{2}} \right )^{2} \right ]$ $\displaystyle \small \left [x^{4}+\frac{1}{x^{4}} \right ]$
⇒ $\displaystyle \small \left (x^{2}-\frac{1}{x^{2}} \right )\left (x^{2}+\frac{1}{x^{2}} \right )$ $\displaystyle \small \left [x^{4}+\frac{1}{x^{4}} \right ]$
⇒ $\displaystyle \small \left (x-\frac{1}{x} \right )\left (x+\frac{1}{x} \right )$ $\displaystyle \small \left (x^{2}+\frac{1}{x^{2}} \right )\left (x^{4}+\frac{1}{x^{4}} \right )$
Equations
1. Simple Equation
Equation consists of one unknown quantity.
2. Equation with Two Variables
Equation of first degree consists of two unknown quantities.
Quadratic Equation
Consider the equation,
$\displaystyle \small ax^{2}+bx+c=0$
The value of 'x' is given by,
$\displaystyle \small x=\frac{-b\pm \sqrt{b^{2}-4ac}}{2a}$
The degree of equation is 2, hence it has two factors say $\displaystyle \small \alpha$ and $\displaystyle \small \beta$
$\displaystyle \small \alpha=\frac{-b+ \sqrt{b^{2}-4ac}}{2a}$
$\displaystyle \small \beta=\frac{-b- \sqrt{b^{2}-4ac}}{2a}$
Note: $\displaystyle \small \alpha +\beta =\frac{-b}{a}$ and $\displaystyle \small \alpha.\beta =\frac{c}{a}$
$\displaystyle \small (+)\times (+)=+$
$\displaystyle \small (+)\times (-)=-$
$\displaystyle \small (-)\times (+)=-$
$\displaystyle \small (-)\times (-)=+$
$\displaystyle \small \frac{(+)}{(+)}=+$
$\displaystyle \small \frac{(+)}{(-)}=-$
$\displaystyle \small \frac{(-)}{(+)}=-$
$\displaystyle \small \frac{(-)}{(-)}=+$
Laws of Indices
1. $\displaystyle \small a^{n}=x$
where,
a = base
n = power
x = value of $\displaystyle \small a^{n}$
2. $\displaystyle \small a^{n}.a^{m}=a^{n+m}$
3. $\displaystyle \small a^{n}\div a^{m}=a^{n-m}$
4. $\displaystyle \small (a^{n})^{m}=a^{n.m}$
5. $\displaystyle \small (a.b)^{n}=a^{n}.b^{n}$
6. $\displaystyle \small a^{-n}=\frac{1}{a^{n}}$
7. $\displaystyle \small a^{0}=1$
8. $\displaystyle \small a^{1}=a$
9. $\displaystyle \small \frac{0}{a}=0$
10. $\displaystyle \small \frac{a}{0}=\infty$
Addition and Subtraction of Algebraic Numbers
A term in an algebraic expression involves letters and/or numbers multiplied together.
Ex: 5x [5 is coefficient and x is variable]
-8xy [-8 is coefficient and xy are varibales]
$\displaystyle \small 7x^{2}-5y$ [7 is coefficient of $\displaystyle \small x^{2}$ and -5 is coefficient of y]
We can only add or subtract like terms (like terms have same variables).
First, group the like terms and add or subtract their coefficients.
(i) Simplify 13x + 7y − 2x + 6a
group like terms,
⇒ (13x-2x)+7y+6a
⇒ 11x+7y+6a
(ii) Add (7x+2) and (2x+3)
⇒ 7x+2+2x+3
group like terms,
⇒ (7x+2x)+(2+3)
⇒ 9x+5
(iii) Add $\displaystyle \small (2x^{3}+x^{2}+x)$ and $\displaystyle \small (4x^{3}+2x^{2})$
⇒ $\displaystyle \small 2x^{3}+x^{2}+x+4x^{3}+2x^{2}$
group like terms,
⇒ $\displaystyle \small (2x^{3}+4x^{3})+(x^{2}+2x^{2})+x$
⇒ $\displaystyle \small 6x^{3}+3x^{2}+x$
(iv) Subtract $\displaystyle \small (2x^{2}-4x-1)$ from $\displaystyle \small (6x^{2}-3x+5)$
⇒ $\displaystyle \small 6x^{2}-3x+5-(2x^{2}-4x-1)$
⇒ $\displaystyle \small 6x^{2}-3x+5-2x^{2}+4x+1$
⇒ $\displaystyle \small (6x^{2}-2x^{2})+(-3x+4x)+(5+1)$
⇒ $\displaystyle \small 4x^{2}+x+6$
Multiplication and Division of Algebraic Numbers
Multiplication: Multiply the coefficients and multiply the variables using laws of indices
(i) $\displaystyle \small (6xy)\times (-3x^{2}y^{3})$
⇒ $\displaystyle \small \left \{ 6\times (-3) \right \}$ $\displaystyle \small \times \left \{ xy.x^{2}y^{3} \right \}$
⇒ $\displaystyle \small \left \{ -18 \right \}$ $\displaystyle \small \times \left \{ x^{3}y^{4} \right \}$
⇒ $\displaystyle \small -18x^{3}y^{4}$
(ii) $\displaystyle \small (5x^{2}-2y^{3})(2x^{3}+5y^{4})$
⇒ $\displaystyle \small 5x^{2}(2x^{3}+5y^{4})$ $\displaystyle \small -2y^{3}(2x^{3}+5y^{4})$
⇒ $\displaystyle \small 5x^{2}.2x^{3}+5x^{2}.5y^{4}$ $\displaystyle \small -2y^{3}.2x^{3}-2y^{3}.5y^{4}$
⇒ $\displaystyle \small 10x^{5}+25x^{2}y^{4}-4x^{3}y^{3}-10y^{7}$
Division:
Algebraic Formulae
1. $\displaystyle \small (a+b)^{2}=a^{2}+b^{2}+2ab$
2. $\displaystyle \small (a-b)^{2}=a^{2}+b^{2}-2ab$
3. $\displaystyle \small (a+b)^{2}=(a-b)^{2}+4ab$
4. $\displaystyle \small (a-b)^{2}=(a+b)^{2}-4ab$
5. $\displaystyle \small (a+b)(a-b)=a^{2}-b^{2}$
6. $\displaystyle \small (a+b)^{3}=a^{3}+b^{3}+3ab(a+b)$
7. $\displaystyle \small (a-b)^{3}=a^{3}-b^{3}-3ab(a-b)$
8. $\displaystyle \small a^{3}+b^{3}=(a+b)(a^{2}+b^{2}-ab)$
9. $\displaystyle \small a^{3}-b^{3}=(a-b)(a^{2}+b^{2}+ab)$
10. $\displaystyle \small (a+b+c)^{2}=a^{2}+b^{2}$ $\displaystyle \small +c^{2}+2ab+2bc+2ca$
Factors of Trinomial
Ex: $\displaystyle \small x^{8}-\frac{1}{x^{8}}$
⇒ $\displaystyle \small (x^{4})^{2}-\left (\frac{1}{x^{4}} \right )^{2}$
[ $\displaystyle \small a^{2}-b^{2}=(a+b)(a-b)$]
⇒ $\displaystyle \small \left ( x^{4}-\frac{1}{x^{4}} \right )\left ( x^{4}+\frac{1}{x^{4}} \right )$
⇒ $\displaystyle \small \left [(x^{2})^{2}-\left (\frac{1}{x^{2}} \right )^{2} \right ]$ $\displaystyle \small \left [x^{4}+\frac{1}{x^{4}} \right ]$
⇒ $\displaystyle \small \left (x^{2}-\frac{1}{x^{2}} \right )\left (x^{2}+\frac{1}{x^{2}} \right )$ $\displaystyle \small \left [x^{4}+\frac{1}{x^{4}} \right ]$
⇒ $\displaystyle \small \left (x-\frac{1}{x} \right )\left (x+\frac{1}{x} \right )$ $\displaystyle \small \left (x^{2}+\frac{1}{x^{2}} \right )\left (x^{4}+\frac{1}{x^{4}} \right )$
Equations
1. Simple Equation
Equation consists of one unknown quantity.
Equation of first degree consists of two unknown quantities.
Quadratic Equation
Consider the equation,
$\displaystyle \small ax^{2}+bx+c=0$
The value of 'x' is given by,
$\displaystyle \small x=\frac{-b\pm \sqrt{b^{2}-4ac}}{2a}$
The degree of equation is 2, hence it has two factors say $\displaystyle \small \alpha$ and $\displaystyle \small \beta$
$\displaystyle \small \alpha=\frac{-b+ \sqrt{b^{2}-4ac}}{2a}$
$\displaystyle \small \beta=\frac{-b- \sqrt{b^{2}-4ac}}{2a}$
Note: $\displaystyle \small \alpha +\beta =\frac{-b}{a}$ and $\displaystyle \small \alpha.\beta =\frac{c}{a}$
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