Problem 3 - Algebra for College Students

Question

Perform the indicated divisions of polynomials by monomials. $$ \frac{-24 x^{6}+36 x^{8}}{4 x^{2}} $$

Step-by-Step Solution

Answer

The result is \(9x^6 - 6x^4\).

Step 1: Examine the Polynomials

The given expression is \( \frac{-24 x^{6} + 36 x^{8}}{4 x^{2}} \). We need to divide each term in the numerator by the monomial in the denominator separately.

Step 2: Divide the First Term

Divide the first term \(-24x^6\) by \(4x^2\):\[\frac{-24x^6}{4x^2} = -6x^{6-2} = -6x^4\].So, the division of the first term yields \(-6x^4\).

Step 3: Divide the Second Term

Divide the second term \(36x^8\) by \(4x^2\):\[\frac{36x^8}{4x^2} = 9x^{8-2} = 9x^6\].The result of dividing the second term is \(9x^6\).

Step 4: Combine the Results

Combine the results from Steps 2 and 3 to get the final expression:\(-6x^4 + 9x^6\).

Key Concepts

Monomial

A monomial is a single term algebraic expression. It consists of a number, known as a coefficient, and variable(s) which can have whole number exponents. Monomials don't contain addition or subtraction. For example, in the given exercise, the denominator is a monomial given by \(4x^2\). This expression includes coefficients and a variable

• Coefficient: the number 4
• Variable: \(x\)
• Exponent of the variable: 2

Monomials are the building blocks of algebraic expressions, and operations involving them, like multiplication or division, are usually simpler to perform than those involving more complex polynomials. When dividing a polynomial by a monomial, as demonstrated in the exercise, it's important to divide each term in the polynomial separately by the monomial.

Polynomials

Polynomials are expressions made up of many monomials combined through addition or subtraction. They include coefficients, variables, and non-negative integer exponents. In the provided exercise, the expression in the numerator \(-24x^6 + 36x^8\) is a polynomial composed of two terms:

• The first term is \(-24x^6\)
• The second term is \(36x^8\)

When dividing a polynomial by a monomial, you must divide each term of the polynomial separately by the monomial. It is essential to maintain the order of operations and treat each term individually. This method ensures accurate simplification of the expression as seen in the steps followed during division in the solution provided.

Degree of a Polynomial

In polynomials, the degree is crucial as it helps understand the expression's behavior and complexities. The degree of a polynomial is the highest power of the variable in the expression. Looking back at the exercise, the numerator \(-24x^6 + 36x^8\) has two terms with different degrees:

• \(-24x^6\) has a degree of 6
• \(36x^8\) has a degree of 8

Thus, the degree of the entire polynomial is 8, which is the highest among the terms. When dividing each term by the monomial \(4x^2\), the resulting terms have degrees reduced by 2 because of the exponent in the divisor, \(x^2\). This results in terms with new degrees: \(-6x^4\) and \(9x^6\), showing how polynomial division can change the degree of each term but not the degree of the resulting polynomial, which remains the highest degree term after simplification, i.e., 6 here.