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A coefficient is basically defined as an integer which is multiplied with the variable of a single term or the terms of a polynomial. In other words, it refers to a number or quantity placed with a variable. The variables having no number with them are assumed to have 1 as their coefficient.

Poly means ‘many’

Nomial means ‘terms’

A mathematical expression having one or more algebraic terms is called a polynomial. Commonly it is expressed as a sum of several terms having different powers of same variable or variables.

Let us understand with few examples.

In an expression: {ax+bx+z}

Here, is called variable and “a” and “b” are the coefficients and z is called constant.

In an expression: {{x}^{2}}+5 , 1 is the coefficient of {{x}^{2}} .

Now, to identify a coefficient, it should be remembered that it always comes with a variable. In above example of { ax+bx+z, x} is the variable. Hence, ‘a’ and ‘b’ are the coefficient here.

A coefficient can be a real/imaginary, positive/negative or even in the form of fractions or decimals.

For example:

In an expression:

- 7x, 4.7 is the coefficient of variable x
- In \frac{{-1}}{2}y,\,\,\frac{{-1}}{2} is the coefficient of variable y.

- A coefficient is always attached to a variable, whether variable is a single term or a polynomial.
- In case there is no number or numerical factor in a term, the coefficient of the same is considered a 1.
- The value of the variable is never the same and varies with a situation or according to the question.
- The value of a constant is always fixed and cannot be changed.
- A coefficient can never be zero because when we multiply 0 (as a coefficient) with any variable, the value of the same results into 0.
- While adding or subtracting polynomials, coefficients of the same variables can only participate in these arithmetic operations. But there is no such rule for multiplication and division.

For example :

On adding 8{{x}^{2}}+12y+9 ** and 1****7x ^{2 }– 3y + 4**

We get, (8 + 17) x^{2 } + (12 – 3) y + (9 + 4)

The answer becomes **25x ^{2 }+ 9y + 13**

**Find out the numerical Coefficients in the term : \frac{{2xy2}}{7}**

**Solution**: \frac{2}{7} is the coefficient of the above term.

**Find out the numerical coefficients in the following algebraic expression****: 7x**^{2 }– 5y + 8**.**

**Solution**: Here we have three algebraic terms. In the term 7x^{2}, the numerical coefficient of the term 7x^{2 }is 7, -5 the coefficient of y, and 8 is a constant. Therefore, the numerical coefficients are 7and -5.

**Find out the numerical coefficients in the expression: 9{{a}^{2}}-3y+8**

**Solution**: Amongst these three algebraic terms, the numerical coefficient of the term ~9{{a}^{2}} is 9, -3 is the coefficient of y, and 8 is a constant. Therefore, the numerical coefficients are 9 and -3.

**Find out the numerical coefficients in the expression: 7{{w}^{2}}+8e+5**

**Solution**: In the term 7{{w}^{2}} the numerical coefficient of the term 7{{w}^{2}} is 7, 8 is the coefficient of e, and 5 is a constant. Therefore, the numerical coefficients are 7 and 8.

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