6 Algebraic Expressions

You know

• to write the terms, coefficients and factors of an algebraic expression.
• to classify an algebraic expression as monomial, binomial, trinomial.
• to identify like terms.
• to add and subtract algebraic expression.

You will learn

• multiplication and division of given polynomials.
• the difference between an identity and an equation.
• algebraic identities and their applications.
• factorization of algebraic expression by regrouping , by taking common factors or using algebraic identities.

Let us recall the basic definitions of algebra

Constants and variables : A quantity having a fixed numerical value is called a constant whereas variables in algebra are letters such as x, y, z or any other letter that can be used to represent unknown numbers.

Algebraic expression : An expression which has a combination of constants and variables connected to each other by one or more operation (+,-,X,÷) is called an algebraic expression.

Example are all algebraic expressions

Term : The parts of an algebraic expression separated by an addition or a subtraction sign are called terms of the expression. In the expression the terms of the expression are are variable terms as their values will change with the value of x, while (-4) is a constant term.

On the basis of the number of terms in an algebraic expression, they are classified as monomials, binomials, trinomials and polynomials.

Monomials are algebraic expressions having one term .

Binomials are algebraic expressions having two terms.

Trinomials are algebraic expressions having three terms.

Polynomials are algebraic expressions having one or more than one term.

Remember – Only expressions with positive powers of variables are called polynomials. An expression of the type is not a polynomial as and the power of variable p is (- 1) which is not a whole number.

Example 1

Classify the algebraic expressions as monomials, binomials or trinomials.

1. 
2. 
3. 
4. 
5. 

Solution

		binomial
		monomial
		trinomial
		monomial
		binomial

Like and Unlike terms : Terms having the same algebraic factors are called like terms . The numerical coefficients may be different. 2x²yz, 5x²yz, 8x²yz and 2x²yz are like terms

3p ³q², 7p ³q²and 9p ³q² are also like terms.

Unlike terms : Terms having different algebraic factors are called unlike terms, , 3x²yz

3p ³q² are unlike terms.

Addition and Subtraction of Algebraic Expressions.

In algebra, like terms can be added or subtracted.

To add or subtract algebraic expressions we can use the horizontal method or the column method.

The horizontal method

All algebraic expressions are written in a horizontal line; the like terms are then grouped. The sum or difference of the numerical coefficients is then found.

Example 2

Add the following

Solution

Example 3

Subtract

Solution

The column method

In the column method, each expression is written in a separate row in such a way that like terms are arranged one below the other in a column. The sum or difference of the numerical coefficients is then found.

Example 4

Add :

Solution

To add by horizontal method, collect the like terms and add coefficients.

To add by column method, arrange the like terms in column and add

Example 5

Subtract :

Solution

We know that the subtraction of two algebraic expressions or terms is addition of the additive inverse of the second term to the first term. Since the additive inverse of a term has opposite sign of the term, hence we can say that in subtraction of algebraic expressions change + to – and change – to + for the term to be subtracted and then add the two terms

To subtract by column method, arrange the like terms in columns and change the sign of the subtrahend

Example 6

What should be added to to get

Solution

The expression to be added will be

Exercise 6.1

1. Classify the algebraic expressions as monomials, binomials or trinomials. Also write the terms of the expression

2. Add the following algebraic expressions by the horizontal method

1. 
2. 
3. 
4. 

3. Add the following algebraic expressions.

1. 

2. 

3. 

4. 

4. Subtract the following expressions.

1. 
2. 
3. 
4. 

5. Subtract the sum of from the sum of .
6. Two adjacent sides of a rectangle are . What will be the perimeter of the rectangle.
7. The perimeter of a triangle is and the measure of two sides is. What will be the measure of the third side?
8. What should be added to to get .
9. What should be subtracted from to get
10. By how much is greater than .

Multiplication of Algebraic Expressions

Multiplication of a monomial by another monomial

To multiply 2 monomials

1. Multiply the numerical coefficients
2. Multiply the literal coefficients and use laws of exponents if variables are same.

The product of two monomials is always a monomial.

Example 1

Find the product of

1. 
2. 
3. 
4. 

Solution

1. 
2. 
3. 
4. 

Geometrical interpretation of product of two monomials

The area of a rectangle is given by the product of length and breadth.

If we consider the length as l and breadth as b, then

Area of rectangle = l x b

Thus, it can be said that the area of a rectangle is product of two monomials.

Let us consider a rectangle of length 4p and breadth 3p,

Area of rectangle ABCD =AB x AD = 4p x 3p = 12p²

Multiplication of a monomial by a binomial

To multiply a monomial by a binomial, we use the distributive law

1. Multiply the monomial by the first term
2. Multiply the monomial by the second term of the binomial.
3. The result is the sum of the two terms

The product of a monomial and a binomial is always a binomial.

Example 2

Find the product

1. 
2. 
3. 

Solution

1. 
2. 
3. 

Example 3

Multiply

Solution

Geometrical interpretation of product of a monomial and a binomial

Area of rectangle = l x b

Let us draw a rectangle ABCD with length (p+q) and breadth k.

Take a point P on AB such that AP = p and PB = q.

Draw a line parallel to AD from the point P, PQ⫽AD meeting DC at Q.

Area of rectangle ABCD = area of rectangle APQD +area of rectangle PBCQ

= k x p + k x q

= k(p + q)

Thus, the product k(p + q) represents the area of a rectangle with length as a binomial (p+q) and breadth as a monomial k.

Multiplication of a monomial by a polynomial

To multiply a monomial with a binomial, we can extend the distributive law further

The product of a monomial and a polynomial is a polynomial.

Example 3

Find the product of

Solution

We have multiplied horizontally in all the above examples

We can also multiply vertically as shown below

Multiply

Geometrical interpretation of product of a monomial and a polynomial

Let us consider a rectangle with length = (p +q + r) and breadth= k

Take points M and N on AB such that

AM = p and MN = q and NB = r

.from the points M and N draw parallel to AD,

MX⫽AD and NY⫽AD meeting DC at X and Y.

Area of rectangle ABCD = area of rectangle AMXD +area of rectangle MNYX +area of rectangle NBCY

Area of rectangle ABCD=pk + qk + rk = k(p + q+ r)

Thus, the product of a monomial and a polynomial represents the area of a reactangle with length as a polynomial and breadth as a monomial.

Example 4

Simplify

Solution

Multiplication of binomials

To multiply two binomials (a + b) and (c + d) we will again use the distributive law of multiplication over addition twice

Example 5

Multiply

Solution

We have multiplied horizontally in all the above examples

We can also multiply vertically as shown below

Multiplication of polynomial by a polynomial

A polynomial is an algebraic expression having 1 or more than one term

To multiply two polynomials, we will use the distributive property that is multiply each term of the first polynomial with each term of the second polynomial.

Example 6

Multiply

Solution

We have multiplied horizontally in the above example, We can also multiply vertically as shown below

 

Exercise 6.2

Multiply the following monomials
2a and 9b
Find the following products and evaluate for x = 1, y = -1
Find the following products by horizontal method
Find the following products using column method
Find the area of the rectangle with the given measurements
Length = 3p, breadth = 4p	Length = (2a+4), breadth = 5a
Multiply the following
Simplify the following expressions
Multiply .
Simplify
If the length of a rectangle is and breadth is 3abc,find the area of the rectangle.

Algebraic identities

An identity is a special type of equation in which the LHS and the RHS are equal for all values of the variables.

The above equation is true for all possible values of a and b; so it is called an identity.

An identity is different from equation as an equation is not true for all values of variables,;it has a unique solution.

Example

There are a number of identities which are used in mathematics to make calculations easy. We are going to study 4 basic identities

Verification of identities

1. 

2. 

3. 

4. 

in this identity a and b can be positive or negative

Geometrical verification of identities

1. Geometrical demonstration for.

Draw a square with length as shown in the figure.

Let the area of original square be X

then, area of Square PQRS=(side)²

∴ ,

Mark a point M on PQ such that length of PM = a and length of MQ= b.

Draw a line MC parallel to PS intersecting SR at C.

Similarly, mark a point B on RQ such that RB = a and QB = b.

Draw a line BD parallel to QP intersecting PS at D.

The whole square is divided into 2 squares and 2 rectangles say A1, A4,A2and A3

Area of Square X1 = side²= a²

Area of rectangle X2= length x breadth = ab

Area of rectangle X3= length x breadth = ab

Area of Square X4 = side²= b²

area of Square PQRS = sum of inside area = area of X1+ area of X2+ area ofX3+ area ofX4

2. Geometrically demonstration for .

We draw a square with length a as shown in the figure.

Let the area of original square is A

Then, area of Square PQRS=(side)²

∴

Mark a point M on PQ such that the length of PM = a-b and length of MQ= b.

Draw a line MC parallel to PS intersecting SR at C.

Similarly, mark a point B on RQ such that RB = a – b and QB = b.

Draw a line BD parallel to QP intersecting PS at D.

The whole square