Problem 8 - Beginning and Intermediate Algebra

Question

Divide. $$\frac{-15 m^{6}+10 m^{5}+20 m^{4}-35 m^{3}}{5 m^{3}}$$

Step-by-Step Solution

Answer

The short answer is \(-3m^3 + 2m^2 + 4m - 7\).

Step 1: Divide the first term

Divide the first term, \(-15m^6\), by \(5m^3\): \[ \frac{-15m^6}{5m^3} = -3m^3 \]

Step 2: Divide the second term

Divide the second term, \(10m^5\), by \(5m^3\): \[ \frac{10m^5}{5m^3} = 2m^2 \]

Step 3: Divide the third term

Divide the third term, \(20m^4\), by \(5m^3\): \[ \frac{20m^4}{5m^3} = 4m \]

Step 4: Divide the fourth term

Divide the fourth term, \(-35m^3\), by \(5m^3\): \[ \frac{-35m^3}{5m^3} = -7 \]

Step 5: Combine the results

Combine the results from Steps 1-4 to get the quotient: \[ -3m^3 + 2m^2 + 4m - 7 \] So, the final answer is \(-3m^3 + 2m^2 + 4m - 7\).

Key Concepts

Algebraic Expressions

Algebraic expressions are a combination of numbers, variables, and arithmetic operations such as addition, subtraction, multiplication, and division. These expressions are fundamental in algebra and serve as the key to representing mathematical relationships and solving equations.
An algebraic expression might look like this:

• \(3x + 5\)
• \(4y^2 - 2y + 7\)
• \(-15m^6 + 10m^5 + 20m^4 - 35m^3\) (from our original problem)

The goal of working with algebraic expressions is often to simplify or manipulate them to find unknown values or write them in a more convenient form. Understanding algebraic expressions enables students to grasp more complex mathematical concepts and problem-solving techniques.

Monomials

A monomial is the simplest form of an algebraic expression and consists of just a single term. It is made up of a constant, a variable, or the product of both. Monomials do not include addition or subtraction between terms.
Here are a few examples of monomials:

• \(5x\)
• \(-3y\)
• \(8m^{2}\)
• \(-15m^6\) (a term from our exercise)

In the division of polynomials, each term in the polynomial may be treated as a monomial. When dividing monomials, the process involves dividing the coefficients (the numerical parts) and subtracting the exponents of like bases according to the laws of exponents.

Polynomial Simplification

Polynomial simplification involves reducing a polynomial to its simplest form. To achieve this, you perform operations like combining like terms, which are terms in the polynomial that have the same variable and exponent. For example, in the exercise, once each term is divided by the monomial \(5m^3\), the results are gathered and simplified into one expression.
The process involves:

• Breaking down each individual term by the divisor.
• Combining the results of each division to form the simplest polynomial.

For our example, by simplifying, we turned a complex polynomial into the simpler expression: \(-3m^3 + 2m^2 + 4m - 7\). This reduces the polynomial into its most elemental, yet equivalent form, making it easier to comprehend and work with further.

Long Division in Algebra

Long division in algebra is akin to the long division process learned in arithmetic but applied to polynomials. It is a systematic method for dividing a polynomial by another polynomial or a monomial, which is particularly useful when the divisor is not as straightforwardly factorable.
The process of polynomial long division can be broken down into steps akin to numerical long division:

• Divide the leading term of the dividend by the leading term of the divisor.
• Multiply the entire divisor by this result and subtract from the original polynomial.
• Repeat the process with the resulting polynomial until the remainder is of a lesser degree than the divisor.

In our specific exercise, since we were dividing by a monomial, each term could be treated individually, simplifying the process. The long division method is more involved when dividing by a polynomial with multiple terms, requiring careful alignment and organization of each step.