We have learnt already and become familiar with what algebraic expressions (or simply expressions) are. Express or elaborate or expanse more about the thing that we discussing. Examples of expressions are:
x + 3, 2y − 5, 3x2, 4xy + 7 etc.
The Algebraic expression has combination of variable or variables. Here x and y are the variables.
Take the expression 4x + 5. This expression is made up of two terms, 4x and 5. Terms are added to form expressions. Terms themselves can be formed as the product of factors.
The expression 7xy − 5x has two terms 7xy and −5x. The term 7xy is a product of factors 7, x and y. The numerical factor of a term is called its numerical coefficient or simply coefficient. The coefficient in the term 7xy is 7 and the coefficient in the term 5x is −5.
Expression that contains only one term is called a monomial. Expression that contains two terms is called a binomial. An expression containing three terms is a trinomial and so on.
In general, an expression containing, one or more terms with non-zero coefficient (with variables having non negative integers as exponents) is called a polynomial. A polynomial may contain any number of terms, one or more than one.
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Examples |
Expression |
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Monomials |
2x2, 3xy, −7z, 5xy2, 10y, −9, 82mnp, etc |
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Binomials |
a + b, 4l + 5m, a + 4, 5 −3xy, z2 − 4y2, etc. |
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Trinomials |
a + b + c, 2x + 3y − 5, x2y − xy2 + y2, etc. |
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Polynomials |
a + b + c + d, 3xy, 7xyz − 10, 2x + 3y + 7z, etc. |
There could be various expression in algebra, Some of them like as per terms of algebra and some unlike terms.
For example - 7x, 14x, −13x, 5x2, 7y, 7xy, −9y2, −9x2, −5yx
Like terms from these are:
7x, 14x, −13x are like terms.
5x2 and −9x2 are like terms.
Unlike terms from these are:
Why are 7x and 7y not like?
Why are 7x and 7xy not like?
As we make addition and subtraction with the numbers, here we can also learn how to add and subtract algebraic expressions.
For example, to add 7x2 −4x + 5 and 9x −10, we do
7x2 − 4x + 5
+ 9x − 10
7x2 + 5x − 5
Subtraction of a number is the same as addition of its additive inverse.
Thus subtracting −3 is the same as adding +3. Similarly, subtracting 6y is the same as adding − 6y; subtracting − 4y2 is the same as adding 4y2 and so on.
Multiplication in algebra has different than number system. Algebra has different pattern for the multiplication.
Let's if there has one number 3 and another variable a is than the multiplication of those will be 3a.
2x x 5y x 7z = (2x x 5y) x 7z = 10xy x 7z = 70xyz
We first multiply the first two monomials and then multiply the resulting monomial by the third monomial.
Let us multiply the monomial 3x by the binomial 5y + 2, i.e., find 3x x (5y + 2) = ?
And the result will be 3x x (5y + 2) = (3x x 5y) + (3x x 2) = 15xy + 6x.
Another Exampe - :
3p x (4p2 + 5p + 7)
= (3p x 4p2) + (3p x 5p) + (3p x 7)
= 12p3 + 15p2 + 21p
In multiplication of polynomials with polynomials, we should always look for like terms, if any, and combine them.
(3a + 4b) x (2a + 3b)
= 3a x (2a + 3b) + 4b x (2a + 3b)
= (3a x 2a) + (3a x 3b) + (4b x 2a) + (4b x 3b)
= 6a2 + 9ab + 8ba + 12b2
= 6a2 + 17ab + 12b2
An equation is true for only certain values of the variable in it. It is not true for all values of the variable.
There are three identities. These identities are obtained by multiplying a binomial by another binomial.
(a + b)2 = a2 + 2ab + b2
An identity, since the expression on the RHS is obtained from the LHS by actual multiplication.
(a − b)2 = (a − b) (a − b)
= a (a − b) − b (a − b)
We have
= a2 − ab − ba + b2 = a2 − 2ab + b2
or (a − b)2 = a2 −2ab + b2
(a + b) (a − b) = a2 − b2
For many problems on multiplication of binomial expressions and also of numbers, use of the identities gives a simple alternative method of solving them.
For example- (2x + 3y)2
To solve this we use the identity of (a + b)2 = a2 + 2ab + b2 formulae.
(2x + 3y)2
= (2x)2 + 2(2x) (3y) + (3y)2
= 4x2 + 12xy + 9y2