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Coefficient

What is an Algebraic Expression?

A combination of constants and variables connected by the signs of fundamental operations such as addition, subtraction, multiplication and division is called an algebraic expression. For example, 3 x + 5 y – 6 z is an algebraic expression. 

Every single entity in an algebraic expression is called a Term. In other words, various parts of an algebraic expression which are separated by the signs, + or – are called the terms of the expression. For example, 6 x 3 + 5 x 2 y – 8 is an algebraic expression consisting of 3 terms, namely, 6 x 3, 5 x 2 y and 8.

Let us now understand what we mean by factors of an algebraic expression. 

Factors of an Algebraic Expression

We know that each of the terms in an algebraic expression is a product of one or more number ( s ) and  / or literal number (s ). These number ( s )  and  / or literal number (s ) are known as the factors of that term. For example, in the binomial, 4 x y + 9 z, 4 x y and 9 z are two terms. In the term 4 x y, for instance, 4, x and y are its factors. Here, we can clearly see that 4 is the numerical factor while x and y are literal factors.

Now that we have understood the meaning of factors let us now understand what we mean by the coefficient of an algebraic expression.

What is a Coefficient?

In a term of an algebraic expression, any of the factors with the sign of the term is called the coefficient of the product of the other factors. For example, in the monomial, 4 x y, the coefficient of y is 4 x, the coefficient of 4 is 4 y and the coefficient of x y is 3. The variables which do not have a number with them are assumed to be having 1 as their coefficient. For example, in the expression 6 x, 6 is the coefficient but in the expression x 2 + 6, 1 is the coefficient of x2. In other words, we can say that a coefficient is a multiplicative factor in the terms of a polynomial, a series, or an algebraic expression.

Types of Coefficients in an Algebraic Expression

In an algebraic expression, there are two types of coefficients – 

  1. Numerical Coefficients
  2. Literal Coefficients

Let us understand both of them one by one.

Numerical Coefficients

We know that a term is usually formed by the product of a number and one or more other factors. A numerical coefficient is a number that is a factor of the remaining variables in a term. For example, in the term 6 x y z, 6 is the numerical coefficient of x y z.

Literal Coefficients

A literal coefficient is the variable part of a term that is a factor of the numerical part. For example, in the term 6 x y z,  x y z is the literal coefficient of 6.

Properties of a Coefficient

We know now that a coefficient can be positive or negative, real or imaginary, or in the form of decimals or fractions. Further, the properties or characteristics of a coefficient can be described as under – 

  1. A coefficient is always attached to a variable. For example, in the algebraic expression 4 x y + 7, 4 is the numerical coefficient of x y, 4 is the coefficient of 4 y, y is the coefficient of 4 x but 7 is its own coefficient as it is a constant term with no variable. So, the number when alone in an algebraic expression is just a constant.
  2. A variable without a number has 1 as its coefficient. As we have learnt above, in the expression 9 x, 9 is the coefficient but in the expression x 2 + 8, 1 is the coefficient of x 2.
  3. The value of a variable is never the same. It varies according to the question and situation, and therefore, it is called a variable.
  4. The value of a constant is constant because its value is always fixed, and it cannot be changed.

Another important concept when it comes to earning about coefficients is the Leading Coefficient. Let us understand what we mean by this term.

Leading Coefficient in Polynomials

In a polynomial, the leading coefficient is in the term with the highest power of x. this term is called the leading term. If the polynomial is written in decreasing order of powers of x, the leading coefficient will be the first coefficient in the first term. Let us understand it using an example.

Suppose we have the polynomial 3 + 2 x 2 – 4 x 3

Now, in this polynomial, the highest power of x is 3, so the degree is 3. The leading term is the term containing that degree is – 4 x 3. We can clearly see that the numerical part of this term is – 4. Hence, – 4 is the leading coefficient of the given polynomial.

We have learnt the word coefficient and its use in algebraic expressions and polynomials. But the usage of the coefficient is not restricted to only these two areas in mathematics. It has a number of other uses as well. Let us where else in mathematics do we use the word coefficient?

Coefficient in Other Areas of Mathematics

The term “coefficient” is used in many different ways in other fields. For example, in statistics, correlation coefficients tell us whether two sets of data are connected. Similarly, we have reliability coefficient, where they are measures of reliability (e.g. two teachers agreeing on a certain ranking) and agreement (the stability or consistency of match scores). Let us see the terms that are used in correlation coefficient and reliability coefficient.

Correlation Coefficient

  1. The Pearson’s Correlation coefficient – The Pearson’s correlation coefficient tells us the degree of correlation between two variables. It is probably the most widely used correlation coefficient.
  2. The Spearman rank Correlation coefficient  – The Spearman rank correlation coefficient is the nonparametric version of the Pearson correlation coefficient.
  3. The Point Biserial Correlation coefficient – The Point Biserial Correlation coefficient is another special case of Pearson’s correlation coefficient. It measures the relationship between one continuous variable and one naturally binary variable.
  4. The validity coefficient – The validity coefficient tells you how strong or weak your experiment results are.

Reliability Coefficient

  1. The coefficient alpha – The coefficient alpha, also known as Cronbach’s alpha is a way to measure the reliability or internal consistency of a psychometric instrument.
  2. The Intraclass correlation coefficient – The Intraclass correlation coefficient measures the reliability of ratings or measurements for clusters, data that has been collected as groups or sorted into groups.
  3. Test-Retest reliability coefficients – Test-Retest reliability coefficients measure test consistency — the reliability of a test measured over time.

Some other commonly used coefficients include – 

  1. The coefficient of variation – The coefficient of variation tells us how data points are dispersed around the mean.
  2. The gamma coefficient – The gamma coefficient tells us how closely two pairs of data match.
  3. Binomial coefficients – Binomial coefficients  tell us how many ways there are to choose 2 things out of a larger set

Solved Examples

Example 1 Write down the coefficient of x in each of the following – 

a) 3 xb) – 4 a xc) 5 x y 2
d) x y ze) – $\frac{3}{2}$ x + 5f) – $\frac{5}{2}$ x y z 2

Solution There are six parts in the given question where we have to find the coefficient of x. Let us solve each part one by one.

a) We have been given the algebraic expression 3 x 

It is important to note here that 3 x is a single term and there are two parts in this term, namely, 3 and x. Since other than x, the remaining value in the term is 3, therefore, the coefficient of x in the term 3 x is 3.

b) We have been given the algebraic expression – 4 a x

It is important to note here that – 4 a x is a single term and there are three parts in this term, namely, – 4, a and x. Since other than x, the remaining values in the term are -4 and a, therefore, the coefficient of x in the term -4 a x  is -4 a.

c) We have been given the algebraic expression 5 x y 2

It is important to note here that 5 x y 2 is a single term and there are three parts in this term, namely, 5, x and y 2. Since other than x, the remaining values of the term are 5 and y 2, therefore, the coefficient of x in the term 5 x y 2 is 5 y 2.

d) We have been given the algebraic expression x y z

It is important to note here that x y z is a single term and there are three parts in this term, namely, x, y and z. Since other than x, the remaining values of the term are y and z, therefore, the coefficient of x in the term x y z is y z.

e) We have been given the algebraic expression – $\frac{3}{2}$ x + 5

It is important to note here that – $\frac{3}{2}$ x + 5 is a binomial, i,e, it has two terms, namely – 32 x and 5. Now, since we have been asked to find the coefficient of x, we are concerned with only the term that contains x as a variable. In this case that term is – $\frac{3}{2}$ x. Now, – $\frac{3}{2}$ x  is a single term and there are two parts in this term, namely, – $\frac{3}{2}$ and x. Since other than x, the remaining values of the term is – $\frac{3}{2}$ , therefore, the coefficient of x in the term – $\frac{3}{2}$ x is – $\frac{3}{2}$ . Hence, overall, the coefficient x in the algebraic expression – $\frac{3}{2}$ x + 5 is – $\frac{3}{2}$.

f) We have been given the algebraic expression – $\frac{5}{2}$ x y z 2

It is important to note here that – $\frac{5}{2}$ x y z 2is a single term and there are four parts in this term, namely, – $\frac{5}{2}$, x, y and z 2. Since other than x, the remaining values of the term are – $\frac{5}{2}$, y and z 2, therefore, the coefficient of x in the term – $\frac{5}{2}$ x y z 2 is – $\frac{5}{2}$ y z 2.

Example 2 Write the numerical coefficient of each term of the following algebraic expressions.

a) x 2 – 7 x 2 y + 5 x y 2 – 2x

b) -2 a 3 + 7 a b 2 – 6 a b + 8 a

Solution We are required to find the numerical coefficient of each of the terms of the given algebraic expressions. We must first recall that a numerical coefficient is a number that is a factor of the remaining variables in a term. Let us find them one by one.

a) The given algebraic expression is x 2 – 7 x 2 y + 5 x y 2 – 2x

Let us first identify the terms given the algebraic expression. The terms are    x 2,  – 7 x 2 y,  5 x y 2 and – 2x.

Now, let us find the numerical coefficient of each of these terms.

The first term of the algebraic expression is x 2. Recall that we have learnt that the variables which do not have a number with them are assumed to be having 1 as their coefficient. Therefore, the coefficient of x 2 is 1.

Now, the second term of the algebraic expression is – 7 x 2 y. We can clearly see that – 7 is the numerical part of this term. Hence, – 7 is the numerical coefficient of – 7 x 2 y.

Next, the third term of the algebraic expression is 5 x y 2. We can clearly see that 5 is the numerical part of this term. Hence, 5 is the numerical coefficient of 5 x y 2.

The last term of the algebraic expression is – 2x. We can clearly see that – 2 is the numerical part of this term. Hence, – 2 is the numerical coefficient of – 2 x.

b) The given algebraic expression is -2 a 3 + 7 a b 2 – 6 a b + 8 a

Let us first identify the terms given the algebraic expression. The terms are    -2 a 3, 7 a b 2, – 6 a b and 8 a

Now, let us find the numerical coefficient of each of these terms.

The first term of the algebraic expression is 2 a 3. We can clearly see that 2 is the numerical part of this term. Hence, 2 is the numerical coefficient of 2 a 3.

Now, the second term of the algebraic expression is 7 a b 2. We can clearly see that 7 is the numerical part of this term. Hence, 7 is the numerical coefficient of 7 a b 2.

Next, the third term of the algebraic expression is – 6 a b. We can clearly see that – 6 is the numerical part of this term. Hence, – 6 is the numerical coefficient of – 6 a b.

The last term of the algebraic expression is 8 a. We can clearly see that 8 is the numerical part of this term. Hence, 8 is the numerical coefficient of 8 a.

Example 3 What is the coefficient in this algebraic expression 7 d + 2b ?

Solution We have been given the algebraic expression 7 d  + 2b. 

Let us first identify the terms given the algebraic expression. The terms are 

7 d  + 2b 

Now, let us find the numerical coefficient of each of these terms.

The first term of the algebraic expression is 7 d. We can clearly see that 7 is the numerical part of this term. Hence, 7 is the numerical coefficient of 7 d.

Next, the second term of the algebraic expression is 2 b. We can clearly see that 2 is the numerical part of this term. Hence, 2 is the numerical coefficient of 2 b.

Key Facts and Summary

  1. A combination of constants and variables connected by the signs of fundamental operations such as addition, subtraction, multiplication and division is called an algebraic expression.
  2. Various parts of an algebraic expression which are separated by the signs, + or – are called the terms of the expression.
  3. Each terms in an algebraic expression is a product of one or more number ( s ) and  / or literal number (s ). These number ( s )  and  / or literal number (s ) are known as the factors of that term.
  4. In a term of an algebraic expression, any of the factors with the sign of the term is called the coefficient of the product of the other factors.
  5. A numerical coefficient is a number that is a factor of the remaining variables in a term.
  6. There are two types of coefficients in algebraic expression – numerical coefficients and literal coefficients.
  7. A literal coefficient is the variable part of a term that is a factor of the numerical part.
  8. In an algebraic expression, the leading coefficient is in the term with the highest power of x. this term is called the leading term.
  9. Apart from algebraic expressions and polynomials, the term “coefficient” is used in many different ways in other fields.

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