Question

Divide. Divide \(9 c^{2}+3 c\) by \(c\).

Short answer

The result of the division \(9c^2 + 3c\) by \(c\) is \(9c + 3\).

Step-by-step solution

Divide the first term

Divide the first term in the numerator \((9c^2)\) by \(c\). To divide, subtract the exponent of \(c\) in the denominator from the exponent of \(c\) in the numerator. The result is \(9c\) (since \(9c^2 = 9c^{2-1} = 9c\).

Divide the second term

Divide the second term in the numerator \((3c)\) by \(c\). This simply results in the coefficient of the term in the numerator, which is \(3\) (since \(3c = 3c^{1-1} = 3)\).

Combine the Results

The result of the division are two separate terms, \(9c\) and \(3\), that combine to make the full quotient \(9c + 3\).

Key concepts

Exponents

Understanding exponents is crucial when dealing with polynomial division. Exponents indicate how many times a number or variable is multiplied by itself. For example, in an expression like \(c^2\), the number 2 is the exponent, meaning \(c\) is multiplied by itself once: \(c \times c\).

When dividing terms with exponents, subtract the exponent in the denominator from the exponent in the numerator. For instance, if you divide \(c^3\) by \(c\), you subtract 1 from 3, resulting in \(c^{3-1} = c^2\), thereby reducing the exponent by 1. This rule helps simplify algebraic expressions and is a fundamental concept in polynomial division.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and at least one algebraic operation—addition, subtraction, multiplication, division, or exponentiation. The exercise \(9c^2 + 3c\) is an example of an algebraic expression with two terms.

When we talk about dividing algebraic expressions, like in the provided exercise, it's in essence dividing each term of the numerator independently by the denominator. It's important to approach each term separately to avoid confusion and to ensure that the properties of operations are correctly applied. The division of these expressions follows the same arithmetic rules as numerical division, keeping in mind the role of exponents with variables.

Simplifying Algebraic Expressions

Simplifying algebraic expressions can be seen as tidying up math. It involves reducing expressions to their simplest form, making them easier to understand and work with.

When dividing terms like \(9c^2\) by \(c\), we're simplifying the expression. In the exercise, dividing each term by \(c\) effectively cancels out one instance of \(c\) from each term, as seen through the subtraction of exponents. This exercise showcases that simplification often involves eliminating common factors between numerators and denominators and applying the laws of exponents to achieve the most reduced form of an expression. The goal is always to reach a simpler or more aesthetic expression that communicates the same mathematical relationship as the original.