Coefficient

A coefficient is a number or symbol written before a variable in a mathematical expression that indicates how many times the variable is multiplied.

Coefficients can be positive, negative, or zero.

It is a scalar value that indicates the variable's impact on an expression. When a variable in an expression has no written coefficient, it is assumed to be one, because multiplying by 1 does not change its value.

For example, in the given expression 10x + x² + 7, it has two coefficients:

• 10, which is the coefficient of x.
• 1, which is the coefficient of x², as it doesn't have a number with it, we automatically assume it to be one.
• 7 is the constant.

Types of Coefficients

Coefficients are grouped into different types based on their usage in expressions.

1. Numerical Coefficient

A numerical coefficient is the number part of a term that multiplies the variable.

• In 4x, the numerical coefficient is 4
• In −7ab, the numerical coefficient is -7

2. Leading Coefficient

The leading coefficient is the coefficient of the term with the highest degree in a polynomial.

• In 3x³ − 5x² + 2x + 1, the leading coefficient is 3

Properties of Coefficients

• Linearity: Coefficients exhibit linearity, meaning they distribute over addition and subtraction. For example, in (ax + by), the coefficient (a) multiplies (x), and (b) multiplies (y).

• Commutativity: The order of coefficients and variables does not affect the result when multiplying. For example, 2 × y and y × 2 both represent the product of 2 and y.

• Associativity: Coefficients are associative with multiplication. For instance, in (2 × 3x), the result is the same as (3 × 2x), yielding (6x) in both cases.

• Identity Property: Coefficient (1) serves as the identity element in multiplication. Multiplying any variable by (1) leaves the variable unchanged.

• Additive Identity: Adding (0) as a coefficient does not alter the value of the expression. For example, (3x + 0 = 3x).

• Scalar Multiplication: Coefficients can be multiplied by scalars. For example, 2(3x) = 6x.

• Zero Coefficient: A coefficient of (0) nullifies the variable's contribution to the expression. For instance, (0x = 0)

Related Articles:

• Numerical Coefficient
• Polynomial Formula
• Correlation Coefficient

Solved Examples

Example 1: In the expression 5x-2y+3z, what are the coefficients of x, y, and z?

Solution:

In the expression 5x - 2y + 3z, the coefficients are as follows:

• Coefficient of x: Coefficient of x is the number directly multiplied by x, which is 5.
• Coefficient of y: Coefficient of y is the number directly multiplied by y, which is -2. (Note: Coefficients can be negative.)
• Coefficient of z: Coefficient of z is the number directly multiplied by z, which is 3.

So, coefficients of x, y, and z are 5, -2, and 3 respectively.

Example 2: A company produces two types of products, A and B. The profit from selling each unit of product A is $3, and the profit from selling each unit of product B is $5. If the company sells x units of product A and y units of product B, write an expression to represent the total profit.

Solution:

To represent the total profit, we need to multiply the number of units sold for each product by their respective profits and then sum the results.

Here's the expression:
Total profit = (3x + 5y)

Expression represents the profit from selling (x) units of product A, each yielding $3 profit, and (y) units of product B, each yielding $5 profit.

Suppose the company sells 10 units of product A (x = 10) and 15 units of product B (y = 15).

Putting these values into the expression:
Total profit = (3 × 10 + 5 × 15)
= (30 + 75)
= 105

So, if the company sells 10 units of product A and 15 units of product B, the total profit would be $105.

Example 3: Solve the equation 2x + 4 = 10 to find the value of x.

Solution:

To solve the equation 2x + 4 = 10 for x, follow these steps:

Isolate the variable term: Subtract 4 from both sides of the equation to isolate the term containing x:
2x + 4 − 4 = 10−4
2x = 6

Solve for x: Divide both sides by 2 to solve for x:
2x/3 = 6/2
x = 3

So, the value of x that satisfies the equation 2x + 4 = 10 is x = 3.

Practice Questions

Question 1: The perimeter of a rectangle is 10x + 6, where x represents the length of one side of the rectangle. If the width of the rectangle is 2x, find the expression for the length.

Question 2: Factor the expression 4x² + 12x completely.

Question 3: Temperature T in degrees Celsius is given by the formula T = 5x + 32, where x is the temperature in degrees Fahrenheit. If the temperature outside is 20°F, what is the corresponding temperature in degrees Celsius?

Question 4: Evaluate the expression 2x³ - 3x² + x - 4 for x = 2.

Question 5: A charity organization collects donations from two sources: individuals and corporations. For every dollar donated by an individual, the charity receives $0.75, and for every dollar donated by a corporation, the charity receives $0.90. If x represents the amount donated by individuals and y represents the amount donated by corporations, write an expression to represent the total amount received by the charity.