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 x

^{4}+x

^{3}-2x

^{2}+x+1 when divided by x-1.

**Sol:**

p(x)=x

^{4}+x

^{3}-2x

^{2}+x+1

Here a=1

put x=1

p(1)=1

^{4}+1

^{3}-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 x^{3}+3x^{2}+5x+6**Sol:**

p(x)= x^{3}+3x^{2}+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 x^{3}+3x^{2}+5x+6

Read More

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