Coefficient

In mathematics, a coefficient is a number or any symbol representing a constant value that is multiplied by the variable of a single term or the terms of a polynomial. It is usually a number, but sometimes may be replaced by a letter in an expression. For example, in the expression: ax² + bx + c, x is the variable and 'a' and 'b' are the coefficients.

A coefficient refers to a number or quantity placed with a variable. It is usually an integer that is multiplied by the variable and written next to it. The variables which do not have a number with them are assumed to be having 1 as their coefficient. For example, in the expression 3x, 3 is the coefficient of x but in the expression x² + 3, 1 is the coefficient of x². In other words, a coefficient is a multiplicative factor in the terms of a polynomial, a series, or any expression. Observe the following expression which shows that 5 is the coefficient of x² and 8 is the coefficient of y.

A coefficient can be positive or negative, real or imaginary, or in the form of decimals or fractions. Another definition of coefficient says, “Any number with which we multiply a variable." For example, in the term 9.3x, 9.3 is the coefficient of the variable x, and in -5z, -5 is the coefficient.

The coefficient of a variable is the value of the integer or any letter that is present with the variable. For example, the coefficient of variable x in the expression 2x + 3y is 2, and in the same expression, the coefficient of variable y is 3. Similarly, the coefficient of the variable x² in the quadratic expression ax² + bx + c is a.

To find a coefficient of a variable in a term, follow the steps given below:

• Step 1: Encircle the variable along with its power whose coefficient we are finding.
• Step 2: Leave that variable and consider all other numbers or variables written with it. That will be the coefficient.

For example, let us find the coefficients of x and y in the term 5xy. To find the coefficient of x, we can encircle it or underline it. Then, take everything else except for x, i.e. 5y. So, the coefficient of x in the term 5xy is 5y. Similarly, the coefficient of y in the term 5xy is 5x.

In order to find a coefficient, the most important thing that should be remembered is that it always comes with a variable. Let us understand this with the help of the example: 5x²+2y-7. In this expression, we can see that there are three terms: 5x², 2y, and -7. In the first term 5x², x² is the variable and as we know that a coefficient always comes with the variable, so the coefficient is 5. In the second term 2y, y is the variable, so the coefficient is 2. In the third term, -7 is known as the constant, not a coefficient.

A numerical coefficient is a number that is the multiplier of the variables in a term. For example, 3 is the numerical coefficient of the term 3mn. The term numerical coefficient is used for the constant multipliers of the variable. A common example is 4xy. Here, the numerical coefficient of xy is 4.

When a polynomial is written in its standard form, then the coefficient of the term containing the degree or the coefficient written with the first term is called the leading coefficient. In other words, it is the coefficient of the term with the highest power in an expression. Look at the image below showing the leading coefficient in the general form of a polynomial.

Some of the examples of the leading coefficient in polynomials are given below:

• In the expression 4a² - 7a + 9, the leading coefficient is 4.
• In 3xy - 9x + 4, the leading coefficient is 3.
• In 17 - 3x³ + 5xy + 8x, the leading coefficient is -3.

Tips and Tricks on Coefficient

Keep the following points in mind while working with coefficients:

• A coefficient is always attached to a variable.
• A variable without a number has 1 as its coefficient.
• We cannot have 0 as the coefficient, as the value of the term with 0 as the coefficient is 0.

Thinking Out Of the Box!

• Can a variable be a coefficient?
• Can a fraction be a coefficient?

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What is a Coefficient Meaning in Math?

A coefficient is a number or an alphabet that is multiplied by a variable of a single term or the terms of a polynomial. For example, in the term 7x, 7 is the coefficient. The coefficient of x in the term 3xy is 3y.

How do you Find the Coefficient?

To find the coefficient, we can cover the variable and look for numbers or alphabets present with it. For example, to find the coefficient of m in the term 10mn, we can hide m, and then we are left with 10n which is the required coefficient.

Can a Coefficient be Negative?

Yes, a coefficient can be negative as well. For example, in the term -9x, -9 is the coefficient.

Can a Fraction be a Coefficient?

Yes, a fraction can also be a coefficient. For example, in the expression: (3/4)x + 2, 3/4 is a fraction which is the coefficient of x.

What is the Coefficient of 0?

The coefficient of 0 is 0 itself as 0 can be written as 0 times some variable, say, 0x. Then the coefficient of x in 0x is 0.

Can a Coefficient be Zero?

A coefficient cannot be zero because when we multiply 0 (as a coefficient) with any variable, the value of the term results in 0. However, a coefficient can be any natural number, negative number, decimals, or fraction.

What is a Numerical Coefficient in Math?

The term numerical coefficient is used for the multipliers of the variable which are in the form of real numbers. A common example is 4xy. Here, the numerical coefficient of xy, x, or y is 4.

What is a Leading Coefficient?

The leading coefficient is defined as the coefficient of the term with the highest power in a polynomial. For example, in the expression 4 + 3x², 3 is the leading coefficient.

What is the Coefficient of x²?

The coefficient of a variable with no numbers or alphabets attached is always 1. So, the coefficient of x² is 1.

What is the Coefficient of a Constant?

The question "coefficient of a constant" is meaning less as there is no topic of coefficient if there is no variable.

What is a Coefficient Example?

A coefficient is defined as the numbers or alphabets attached with a variable in a term. For example, the coefficient of x in the term 5x⁵ is 5, the coefficient of q in 9pq is 9p, etc.