\displaystyle \textbf{Fundamental Concepts of Algebra}

\displaystyle \textbf{Constants}
\displaystyle \text{A quantity having a fixed numerical value is called a constant}.
\displaystyle \text{Examples: }3,\ 5,\ \frac{2}{5},\ -5,\ \sqrt{10}.
\displaystyle \text{The value of a constant does not change}.
\displaystyle \\

\displaystyle \textbf{Variables or Literals}
\displaystyle \text{A symbol that can take different numerical values is called a variable}
\displaystyle \text{or a literal. Variables are generally represented by letters such as}
\displaystyle x,\ y,\ z,\ a,\ b,\text{ etc.}
\displaystyle \text{Example: If }x=3y,\text{ the value of }x\text{ changes as the value of }y\text{ changes}.
\displaystyle \\

\displaystyle \textbf{Algebraic Expressions}
\displaystyle \text{A combination of constants and variables connected by mathematical}
\displaystyle \text{operations is called an algebraic expression}.
\displaystyle \text{Examples: }3x+7,\quad 5x^2-2xy+4y,\quad \frac{x}{3}+2y.
\displaystyle \\

\displaystyle \textbf{Terms of an Algebraic Expression}
\displaystyle \text{The parts of an algebraic expression separated by }+\text{ or }-\text{ signs}
\displaystyle \text{are called the terms of the expression}.
\displaystyle 3x+7y\text{ contains two terms: }3x\text{ and }7y.
\displaystyle 3x^2+2y^3-3xy\text{ contains three terms: }3x^2,\ 2y^3,\text{ and }-3xy.
\displaystyle \text{The sign preceding a term is considered a part of that term}.
\displaystyle \\

\displaystyle \textbf{Various Types of Algebraic Expressions}

\displaystyle \textbf{Monomial}
\displaystyle \text{An algebraic expression containing only one term is called a monomial}.
\displaystyle \text{Examples: }3x,\quad5y,\quad-2x,\quad y^3,\quad-\frac{4}{9}xy.
\displaystyle \\

\displaystyle \textbf{Binomial}
\displaystyle \text{An algebraic expression containing exactly two terms is called a binomial}.
\displaystyle 3x+7y\text{ is a binomial with terms }3x\text{ and }7y.
\displaystyle x^2y+\frac{2z}{3}\text{ is a binomial with terms }x^2y\text{ and }\frac{2z}{3}.
\displaystyle \\

\displaystyle \textbf{Trinomial}
\displaystyle \text{An algebraic expression containing exactly three terms is called a trinomial}.
\displaystyle 3x+7y+3z\text{ has the terms }3x,\ 7y,\text{ and }3z.
\displaystyle x^2y+\frac{2z}{3}-2abc\text{ has the terms }x^2y,\ \frac{2z}{3},\text{ and }-2abc.
\displaystyle \\

\displaystyle \textbf{Multinomial}
\displaystyle \text{An algebraic expression containing two or more terms is called a multinomial}.
\displaystyle \text{Thus, binomials and trinomials are also multinomials}.
\displaystyle 3x+7y+3z\text{ is a multinomial containing three terms}.
\displaystyle x^2y+\frac{2z}{3}-2abc+c^2\text{ is a multinomial containing four terms}.
\displaystyle \\

\displaystyle \textbf{Factors of a Term}
\displaystyle \text{When two or more quantities are multiplied to form a term, each of these}
\displaystyle \text{quantities is called a factor of the term}.
\displaystyle \text{For example, in }-9xy,\text{ the factors are }-9,\ x,\text{ and }y.
\displaystyle \text{A constant factor is called a numerical factor, while a variable factor}
\displaystyle \text{is called a literal factor}.
\displaystyle \\

\displaystyle \textbf{Constant Term}
\displaystyle \text{A term in an algebraic expression that contains no variable is called}
\displaystyle \text{the constant term}.
\displaystyle \text{In }3x+7y+3z-2,\text{ the constant term is }-2.
\displaystyle \text{In }x^2y+\frac{2z}{3}-2abc+c^2+21,\text{ the constant term is }21.
\displaystyle \\

\displaystyle \textbf{Coefficients}
\displaystyle \text{Any factor of a term is called the coefficient of the product of the}
\displaystyle \text{remaining factors}.
\displaystyle \text{The numerical factor of a term is called its numerical coefficient}.
\displaystyle \text{The variable factor of a term is called its literal coefficient}.
\displaystyle \text{For example, in }-9xy,\text{ the numerical coefficient is }-9\text{ and the}
\displaystyle \text{literal coefficient is }xy.
\displaystyle \text{The coefficient of }x\text{ in }-9xy\text{ is }-9y,\text{ and the coefficient of }y
\displaystyle \text{is }-9x.
\displaystyle \text{If no numerical coefficient is written, it is understood to be }1\text{ or }-1.
\displaystyle \text{For example, the numerical coefficients of }x\text{ and }-y\text{ are }1\text{ and }-1.
\displaystyle \\

\displaystyle \textbf{Like and Unlike Terms}
\displaystyle \text{Terms having the same variables raised to the same powers are called}
\displaystyle \text{like terms. Otherwise, they are called unlike terms}.
\displaystyle 7x^2,\quad-7x^2,\quad-\frac{3}{2}x^2\text{ are like terms}.
\displaystyle 7x^2y,\quad-7z^2,\quad-\frac{3}{2}xz^2\text{ are unlike terms}.
\displaystyle \text{Only like terms can be added or subtracted directly}.
\displaystyle \\

\displaystyle \textbf{Polynomials}
\displaystyle \text{An algebraic expression in which the powers of all variables are}
\displaystyle \text{non-negative integers is called a polynomial}.
\displaystyle \text{A variable cannot have a negative or fractional power, nor can a}
\displaystyle \text{variable occur in the denominator of a polynomial}.
\displaystyle 3x+7y+3z-2\text{ is a polynomial}.
\displaystyle x^2y+\frac{2z}{3}-2abc+c^2+21\text{ is also a polynomial}.
\displaystyle 3x+\frac{7}{x}\text{ is not a polynomial because }\frac{7}{x}=7x^{-1}.
\displaystyle \sqrt{x}+2\text{ is not a polynomial because }\sqrt{x}=x^{\frac{1}{2}}.
\displaystyle \\

\displaystyle \textbf{Degree of a Polynomial in One Variable}
\displaystyle \text{The highest power of the variable with a non-zero coefficient is called}
\displaystyle \text{the degree of a polynomial in one variable}.
\displaystyle 3x+7x^2+3x^3-2\text{ is of degree }3\text{ since the highest power of }x\text{ is }3.
\displaystyle 3y-4y^2+3y^5\text{ is of degree }5\text{ since the highest power of }y\text{ is }5.
\displaystyle 3x+\frac{7}{x}\text{ is not a polynomial since }\frac{7}{x}=7x^{-1}.
\displaystyle \\

\displaystyle \textbf{Linear Polynomial}
\displaystyle \text{A polynomial of degree }1\text{ is called a linear polynomial}.
\displaystyle \text{Example: }3x+7\text{ is a linear polynomial of degree }1.
\displaystyle \\

\displaystyle \textbf{Quadratic Polynomial}
\displaystyle \text{A polynomial of degree }2\text{ is called a quadratic polynomial}.
\displaystyle \text{Example: }7x^2+3x+1\text{ is a quadratic polynomial of degree }2.
\displaystyle \\

\displaystyle \textbf{Cubic Polynomial}
\displaystyle \text{A polynomial of degree }3\text{ is called a cubic polynomial}.
\displaystyle \text{Example: }7x^3+3x+1\text{ is a cubic polynomial of degree }3.
\displaystyle \\

\displaystyle \textbf{Constant Polynomial}
\displaystyle \text{A polynomial consisting of a non-zero constant is called a constant}
\displaystyle \text{polynomial. The degree of a non-zero constant polynomial is }0.
\displaystyle 7,\quad4,\quad45,\quad-23,\quad-\frac{2}{3}\text{ are constant polynomials}.
\displaystyle \text{Each of these has degree }0.
\displaystyle \text{The degree of the zero polynomial is not defined}.
\displaystyle \\

\displaystyle \textbf{Degree of a Polynomial in Two or More Variables}
\displaystyle \text{The degree of a term is the sum of the powers of all the variables in}
\displaystyle \text{that term}.
\displaystyle \text{The degree of a polynomial is the highest degree among all its terms}.
\displaystyle \text{Consider }x^3y^2+x^2-2x^2y^4+\frac{2}{3}xy^3z^4.
\displaystyle \text{Degrees of the terms are }3+2=5,\quad2,\quad2+4=6,\quad1+3+4=8.
\displaystyle \therefore \text{the degree of the polynomial is }8.
\displaystyle \\

\displaystyle \textbf{Substitution}
\displaystyle \text{The process of replacing variables in an expression by given values is}
\displaystyle \text{called substitution}.
\displaystyle \text{Example: Consider the expression }3x+7.
\displaystyle \text{If }x=3,\text{ then }3x+7=3(3)+7=9+7=16.
\displaystyle \therefore \text{the value of the expression at }x=3\text{ is }16.
\displaystyle \\

\displaystyle \textbf{Addition of Like Terms}
\displaystyle \text{The sum of two or more like terms is a like term whose numerical}
\displaystyle \text{coefficient is the sum of the numerical coefficients of the given terms}.
\displaystyle \text{Example: }2xy+4xy-xy=(2+4-1)xy=5xy.
\displaystyle \text{Thus, like terms can be combined by adding their numerical coefficients}.
\displaystyle \text{Unlike terms cannot be combined in this manner}.
\displaystyle \\

\displaystyle \textbf{Subtraction of Like Terms}
\displaystyle \text{The difference of two like terms is a like term whose numerical}
\displaystyle \text{coefficient is the difference of their numerical coefficients}.
\displaystyle \text{Example: }7xy-3xy=(7-3)xy=4xy.
\displaystyle \text{Also, }-7xy-4xy=(-7-4)xy=-11xy.
\displaystyle \text{Subtraction can also be performed by adding the additive inverse}.
\displaystyle \text{For example, to subtract }3xy\text{ from }7xy,\text{ change }3xy\text{ to }-3xy.
\displaystyle 7xy-3xy=7xy+(-3xy)=4xy.
\displaystyle \\

\displaystyle \textbf{Addition of Polynomials}
\displaystyle \text{Two or more polynomials can be added by arranging the like terms in}
\displaystyle \text{the same vertical columns and then adding the terms in each column}.
\displaystyle \text{A term occurring in only one polynomial is written in its proper column}.
\displaystyle \text{Example: Add }4x-7y+z+10,\quad5x-3y-28,\quad x+11y+3a+15.

\displaystyle \begin{array}{rrrr} 4x&-7y&+z&\phantom{+3a}+10\\ 5x&-3y&&-28\\ (+)~x&+11y&+3a&+15\\ \hline 10x&+y&+z+3a&-3 \end{array}

\displaystyle \therefore \text{the sum is }10x+y+z+3a-3.
\displaystyle \\

\displaystyle \textbf{Subtraction of Polynomials}
\displaystyle \text{To subtract one polynomial from another, change the sign of every term}
\displaystyle \text{of the polynomial being subtracted and then add the two polynomials}.
\displaystyle \text{Example: Subtract }4x-7y+z+10\text{ from }5x-3y-28.
\displaystyle \text{Changing the signs of the polynomial being subtracted gives}
\displaystyle -4x+7y-z-10.

\displaystyle \begin{array}{rrrr} 5x&-3y&&-28\\ (+)~-4x&+7y&-z&-10\\ \hline x&+4y&-z&-38 \end{array}

\displaystyle \therefore \text{the difference is }x+4y-z-38.
\displaystyle \\

\displaystyle \textbf{Multiplication of Polynomials}

\displaystyle \textbf{Multiplication of Monomials}
\displaystyle \text{To multiply monomials, multiply their numerical coefficients and then}
\displaystyle \text{multiply their variable parts using the laws of exponents}.
\displaystyle \text{Product of monomials}= \text{product of coefficients}\times\text{product of variable parts}.
\displaystyle \text{Example: Multiply }3xy\text{ and }2x.
\displaystyle (3xy)(2x)=(3\times2)(x\times x)y=6x^2y.
\displaystyle \text{Recall that }x^m\times x^n=x^{m+n}.
\displaystyle \\

\displaystyle \textbf{Multiplication of a Polynomial by a Monomial}
\displaystyle \text{Use the distributive law to multiply every term of the polynomial by}
\displaystyle \text{the monomial}.
\displaystyle a(b+c)=ab+ac,\qquad a(b-c)=ab-ac.
\displaystyle \text{Similarly, }(a+b)c=ac+bc,\qquad(a-b)c=ac-bc.
\displaystyle \text{Example: Multiply }(2x+3y)\text{ by }3x^2y.
\displaystyle (2x+3y)(3x^2y) =(2x)(3x^2y)+(3y)(3x^2y).
\displaystyle =6x^3y+9x^2y^2.
\displaystyle \\

\displaystyle \textbf{Multiplication of a Polynomial by a Polynomial}
\displaystyle \text{To multiply two polynomials, multiply every term of one polynomial by}
\displaystyle \text{every term of the other polynomial and then combine the like terms}.
\displaystyle \text{Example: Multiply }(3x^2+2x-5)\text{ by }(2x^2+x).
\displaystyle (3x^2+2x-5)(2x^2+x).
\displaystyle \text{Multiplying by }2x^2\text{ gives }6x^4+4x^3-10x^2.
\displaystyle \text{Multiplying by }x\text{ gives }3x^3+2x^2-5x.

\displaystyle \begin{array}{rrrr} 6x^4&+4x^3&-10x^2&\\ &+3x^3&+2x^2&-5x\\ \hline 6x^4&+7x^3&-8x^2&-5x \end{array}

\displaystyle \therefore (3x^2+2x-5)(2x^2+x) =6x^4+7x^3-8x^2-5x.
\displaystyle \\

\displaystyle \textbf{Division of Polynomials}

\displaystyle \textbf{Division of a Monomial by a Monomial}
\displaystyle \text{Divide the numerical coefficients and use the laws of exponents to}
\displaystyle \text{divide the variable parts}.
\displaystyle \text{Recall that }\frac{x^m}{x^n}=x^{m-n},\quad x\ne0.
\displaystyle \text{Example: Divide }25x^2y^3\text{ by }5xy.
\displaystyle \frac{25x^2y^3}{5xy} =5x^{2-1}y^{3-1}=5xy^2.
\displaystyle \\

\displaystyle \textbf{Division of a Polynomial by a Monomial}
\displaystyle \text{Divide each term of the polynomial separately by the monomial}.
\displaystyle \text{Example: Divide }25x^2y^3+5xy^2\text{ by }5xy.
\displaystyle \frac{25x^2y^3+5xy^2}{5xy} =\frac{25x^2y^3}{5xy}+\frac{5xy^2}{5xy}.
\displaystyle =5xy^2+y.
\displaystyle \\

\displaystyle \textbf{Division of a Polynomial by a Polynomial}
\displaystyle \text{Polynomial long division is used when a polynomial is divided by another}
\displaystyle \text{polynomial. The terms should first be arranged in descending powers}.
\displaystyle \textbf{Step 1: }\text{Arrange the dividend and divisor in descending powers}.
\displaystyle \textbf{Step 2: }\text{Divide the first term of the dividend by the first term}
\displaystyle \text{of the divisor to obtain the first term of the quotient}.
\displaystyle \textbf{Step 3: }\text{Multiply the entire divisor by this quotient term and}
\displaystyle \text{subtract the resulting polynomial from the dividend}.
\displaystyle \textbf{Step 4: }\text{Repeat the process with the remainder until its degree}
\displaystyle \text{is less than the degree of the divisor}.
\displaystyle \text{Example: Divide }x^2-9x-10\text{ by }x+1.
\displaystyle \text{First, }\frac{x^2}{x}=x.
\displaystyle \text{Therefore, the first term of the quotient is }x.
\displaystyle x(x+1)=x^2+x.
\displaystyle (x^2-9x-10)-(x^2+x)=-10x-10.
\displaystyle \text{Next, }\frac{-10x}{x}=-10.
\displaystyle -10(x+1)=-10x-10.
\displaystyle (-10x-10)-(-10x-10)=0.
\displaystyle \therefore \frac{x^2-9x-10}{x+1}=x-10.
\displaystyle \text{Thus, the quotient is }x-10\text{ and the remainder is }0.
\displaystyle \\

\displaystyle \textbf{Simplification by Removal of Brackets and Use of BODMAS}
\displaystyle \text{The BODMAS rule also applies while simplifying algebraic expressions}.
\displaystyle \text{If a }+\text{ sign occurs before a bracket, the bracket can be removed}
\displaystyle \text{without changing the signs of the terms inside it}.
\displaystyle +(a+b-c)=a+b-c.
\displaystyle \text{If a }-\text{ sign occurs before a bracket, the sign of every term inside}
\displaystyle \text{the bracket must be reversed when the bracket is removed}.
\displaystyle -(a+b-c)=-a-b+c.
\displaystyle \text{Example: }5x+(3x-2y)=5x+3x-2y=8x-2y.
\displaystyle \text{Example: }5x-(3x-2y)=5x-3x+2y=2x+2y.
\displaystyle \\

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