Algebra concept deals with letters of unknown values in Algebraic expressions. In solving to find the value of these letters we use different methods, tactics, rules, and formulas. To master these tricks we need to study algebraic expressions and their properties.
Algebraic Expressions
These are the expressions formed with variables and constants.
Eg: 9x+3
Variable In Algebraic Expressions
It is a value that is unknown or keeps on changing.
letters x, y. z, l, m, n………etc are used to denote variable.
Constant In Algebraic Expressions
It is a value that does not change.
eg: 1, 2, …….etc
Coefficient In Algebraic Expressions
It is the value multiplied by a variable.
eg: 3x, 5z here 3, 5 are coefficients of x and z respectively.
Term
The term is defined as a product of constants and one or more variables.
eg: 9x²-5xy
Factors Of A Term In Algebraic Expressions
Let us understand this by considering the term 9x².
9x² can be written as 9×x×x. Here 9, x and x are the factors of term 9x².
Like Terms
The terms which have the same variables as algebraic factors but need not have some numerical are called like terms.
Eg: 9x, -3x
Here in two terms 9x, -3x the variable x is the same but 9, -3 its coefficients numerical values are different. These terms are like terms.
Unlike Terms
The terms which have different variables as algebraic factors then they are unlike terms.
Eg: 9x², -3x
Here variable algebraic factor x² and x are different. so these terms are unlike terms.
Types Of Algebraic Expressions
Monomial
An algebraic expression that has only one term is known as a monomial.
Eg: 9xy, -pl, 7z², 9 etc
Binomial
An algebraic expression with two different terms is called Binomial.
Eg: a²+b², a+5
Trinomial
An algebraic expression with three different terms is called trinomial.
Eg: 9y²+5y+7, x+y+9
Polynomial
An algebraic expression that has one or more terms is called polynomial.
Monomial, binomial and trinomial are all polynomials.
For a polynomial all the powers are positive. It should not have any negative or fractional form powers except positive integers.
General Expression For Polynomials
\begin{align*}
p\left( x\right) =a_{0}x^{n}+a_{1}x^{n-1}+\ldots +a_{n} \\
Here \ \begin{aligned}a_{0}
\neq 0\\
\ \ n= degree \ of \ polynomial\\
n\neq negative \\
x=variable\end{aligned}
\end{align*}
Zero Of A Ploynomial
These are the values of variables that makes the polynomial equal to zero is called zero of a polynomial.
Eg: p(x)=3x-9 for p(k)=0
put x=k
p(k)=3k-9=0 =>3k=9
k=3 is called zero of a polynomial.
Degree Of Algebraic Equation/ Polynomial
In an algebraic equation or plynomial the highest degree among the degress of different terms is called degree of algebraic equation/ polynomial.
Here degree is the sum of exponents of variables and the exponent values are non-negative integers.
Eg: 9x²y+4y-5
This equation has 3 terms 9x²y, 4y and -5
Degree of first term is 2+1=3
Degree of second term is 1
Here 3 is the highest degree among degress of other two terms.
Therefore, Degree of this polynomial is 3.
Identity
An Equality which is satisfied by any value that replaces its variables then the equation is called an identity.
Some Standard Identities In Algebra
Remainder Therom
If p(x) is a polynomial with degree ≥ 1, when this polynomial divided by a linear polynomial x-a, remainder is p(a).
Eg:
Find remainder for x4+x3-2x2+x+1 when divided by x-1.
Sol:
p(x)=x4+x3-2x2+x+1
Here a=1
put x=1
p(1)=14+13-2(1)2+1+1 ⇒1+1-2+1+1⇒2
p(1)=2
Therefore, remainder is 2
Factor Theorm
x-a is a factor of polynomial p(x), if p(a)=0 similarly if x-a is factor of p(x) polynomial, then p(a)=0
Eg:
Check whether 4x+8 is a factor of x3+3x2+5x+6
Sol:
p(x)= x3+3x2+5x+6
let 4x+8 =0 then a=x=-2
Put x=-2 in the above equation
p(-2)= (-2)3+3(-2)2+5(-2)+6 ⇒ -8+12-10+6 ⇒ 0
⇒ p(-2)=0
Here the value of x satisfies the equation for solving it to zero.
Therefore 4x+8 is a factor of x3+3x2+5x+6
Read More
| Algebraic Properties | Algebra Multiplication Rules |
| Algebraic Equations List | AP and GP |
