Category : 8th Class
Algebraic Expressions and Identities
e.g., \[2-3x+5{{x}^{-2}}{{y}^{-1}}+\frac{x}{3{{y}^{3}}}\]
An algebraic expression in which the variables involved have only non-negative integral powers is called a polynomial.
e.g., \[2-3x+5{{x}^{2}}{{y}^{-1}}-\frac{x}{3}x{{y}^{3}}\]
terms. The coefficients of like terms need not be the same.
In other words, terms with the same variables and which have the same exponent are called like or similar terms, otherwise they are called unlike (or) dissimilar terms.
e.g., (1)\[~3{{x}^{3}},\frac{1}{2}{{x}^{3}},- 9x3,.....\]
etc, are like terms.
(2) \[{{\operatorname{x}}^{2}}y,3x{{y}^{2}},-4{{x}^{3}}, .....\]
etc, are unlike terms.
e.g.,
\[5{{x}^{3}}-7x+\frac{3}{2}\]
is a polynomial in\['x'\]of degree 3.
e.g.,
\[5{{x}^{3}}-2{{x}^{2}}{{y}^{2}}3{{x}^{2}}+9y\]
is a polynomial of degree 4 in \['x'\] and 'y'.
(i) Monomial: A polynomial containing 1 term is called a monomial.
(ii) Binomial: A polynomial containing 2 terms is called a binomial.
(iii) Trinomial: A polynomial containing 3 terms is called a trinomial.
(i) A monomial multiplied by a monomial always gives a monomial.
(ii) While multiplying a polynomial by monomial, we multiply every term in the polynomial by the monomial.
(iii) In carrying out the multiplication of a polynomial by a binomial (or trinomial), we multiply term by term, i.e., every term of the polynomial is multiplied by every term in the binomial (or trinomial). Note that in such multiplication, we may get terms in the product which are like and have to be combined.
(i) \[{{\left( a+b \right)}^{2}}={{a}^{2}}+2ab+{{b}^{2}}\]
(ii)\[{{\left( a-b \right)}^{2}}={{a}^{2}}-2ab+{{b}^{2}}\]
(iii) \[\left( a+b \right)\left( a-b \right)={{a}^{2}}-{{b}^{2}}\]
(iv) \[\left( x+a \right)\left( x+b \right)={{x}^{2}}+\left( a+b \right)x+ab\]
(v) These identities are useful in computing squares and products of algebraic expressions. They are alternative methods to calculate products of numbers too.
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