Coefficients

We have learnt how to write a term as a of factors. One of these factors may be numerical and the others algebraic (i.e., they contain variables). The numerical factor is said to be the numerical coefficient or simply the coefficient of the term.

It is also said to be the coefficient of the rest of the term (which is obviously the product of algebraic factors of the term). Thus, in 5xy, 5 is the coefficient of the term. It is also the coefficient of xy. In the term 10xyz, 10 is the coefficient of xyz, in the term −7 x2 y2 –7 is the coefficient of x2 y2.

Try these

(i) 8y + 3x2

8y + 3x2 The terms in the expression are

8y and 3x2

8y + 3x2 is obtained by adding the product of 8 and y with the product of 3 x and x

(ii) 7mn-4

7mn-4 has two terms 7mn and 4 and the expression is obtained by subtracting 4 from the product of 7,m and n. The tree diagram is shown in the next slide.

(iii) 2x2y

2x2y has only one term, that is the expression itself.

The expression is formed by multiplying 4 terms 2, x, x and y. The tree diagrams are shown in the next slide

When the coefficient of a term is +1, it is usually omitted. For example, 1x is written as x; 1x2, 1y2 is written as x2, y2 respectively and so on.

Similarly, the coefficient −1 is indicated only by the minus sign. Thus −1x is written as −x; −1x2 , −1y2 is written as −x2 , −y2 and so on.

Sometimes, the word ‘coefficient’ is used in a more general way. Thus, we say that in the term 5xy, 5 is the coefficient of xy, x is the coefficient of 5y and y is the coefficient of 5x.

In 10xy2 is the coefficient of xy2 , x is the coefficient of 10y2 and y2 is the coefficient of 10x.

Thus, in this more general way, a coefficient may be either a numerical factor or an algebraic factor or a product of two or more factors. It is said to be the coefficient of the product of the remaining factors.

Try these

Identify the coefficientsof the terms of following expressions: 4x – 3y, a + b + 5, 2y + 5, 2xy

(i)4x – 3y,

4x-3y

4x-3y has two terms 4x and -3y.

the coefficient of x is and the coefficient of y is

(ii)a + b + 5,

a+b+5 has 3 terms a,b and a constant that is 5.

the coefficient of a is and b is also . Constant terms have coefficient

(iii)2y + 5

2y+5 has two terms 2y and 5 which is constant.

The coefficient of y is . Constant terms have no coefficient

(iv)2xy

2xy has only one term

which is 2xy and the coefficient of xy is

Example 1

Identify, in the following expressions, terms which are not constants. Give their numerical coefficients:

xy + 4, 13 –y2, 13 – y + 5y2,4p2q - 3pq2 + 5

S. No.	. Expression	Term (which is not a Constant)	Numerical Coefficient
(i)	xy + 4
(ii)	13-y2
(iii)	13 – y + 5y2	– y + 5y2	and
(iv)	4p2q -3pq2+5	4p2q -3pq2	and

S. No.

. Expression

Term (which is not a Constant)

Numerical Coefficient

(i)

xy + 4

(ii)

13-y2

(iii)

13 – y + 5y2

– y + 5y2

and

(iv)

4p2q -3pq2+5

4p2q -3pq2

and

Example 2

(a) What are the coefficients of x in the following expressions? 4x – 3y, 8 – x + y,y2x-y,2z – 5xz

Solution :

S. No.	. Expression	Term with factor x	Coefficient of x
(i)	4x – 3y
(ii)	8 – x + y
(iii)	y2x-y
(iv)	2z – 5xz

S. No.

. Expression

Term with factor x

Coefficient of x

(i)

4x – 3y

(ii)

8 – x + y

(iii)

y2x-y

(iv)

2z – 5xz

(b) What are the coefficients of y in the following expressions? 4x – 3y, 8 + yz, yz2 + 5, my + m

Solution :

S. No.	. Expression	Term with factor y	Coefficient of y
(i)	4x – 3y
(ii)	8 + yz
(iii)	yz2+5
(iv)	my + m

S. No.

. Expression

Term with factor y

Coefficient of y

(i)

4x – 3y

(ii)

8 + yz

(iii)

yz2+5

(iv)

my + m

Algebraic Expressions

Glossary

Coefficients

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Try these

Example 1

Example 2

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Algebraic Expressions

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Glossary

Coefficients

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Try these

Example 1

Example 2