Question

Challenge Problems. $$7 m^{4} n^{2}+7 m^{3} n^{3}-7 m^{2} n^{4} \mathrm{by}\left(-7 m^{2} n^{3}\right)$$

Short answer

After dividing the given polynomial by the monomial, the simplified expression is \(-\frac{m^{2}}{n} - m + n\).

Step-by-step solution

Title - Understanding the Problem

We need to divide the polynomial \(7 m^{4} n^{2} + 7 m^{3} n^{3} - 7 m^{2} n^{4}\) by the monomial \(-7 m^{2} n^{3}\). Division of polynomials by a monomial involves dividing each term of the polynomial by the monomial separately.

Title - Divide the First Term

Divide the first term of the polynomial, \(7 m^{4} n^{2}\), by \(-7 m^{2} n^{3}\). Using the properties of exponents, this results in \(\frac{7 m^{4} n^{2}}{-7 m^{2} n^{3}} = - m^{4-2} n^{2-3} = - m^{2} n^{-1}\). Since negative exponents indicate division by that factor, we write \(- m^{2} n^{-1}\) as \(- \frac{m^{2}}{n}\).

Title - Divide the Second Term

Divide the second term of the polynomial, \(7 m^{3} n^{3}\), by \(-7 m^{2} n^{3}\). This gives us \(\frac{7 m^{3} n^{3}}{-7 m^{2} n^{3}} = - m^{3-2} n^{3-3} = - m n^{0}\). Since any number to the power of zero is 1, the result simplifies to \(-m\).

Title - Divide the Third Term

Divide the third term of the polynomial, \(-7 m^{2} n^{4}\), by \(-7 m^{2} n^{3}\). This results in \(\frac{-7 m^{2} n^{4}}{-7 m^{2} n^{3}} = m^{2-2} n^{4-3} = n^{1}\) or simply \(n\) as \(m^{2-2}\) cancels out to 1.

Title - Combine the Results

After dividing each term by \(-7 m^{2} n^{3}\), combine the results of each division to get the final simplified expression: \(-\frac{m^{2}}{n} - m + n\).

Key concepts

Monomial Division

Polynomial division by a monomial is a process you will often encounter in algebra. It involves simplifying an expression by dividing each term of the polynomial by the same monomial. Let's start with breaking down what this means. A monomial is a single term algebraic expression, such as \(7x^3\) or \(5y\). When you're given a polynomial, for example \(7 m^{4} n^{2} + 7 m^{3} n^{3} - 7 m^{2} n^{4}\), and asked to divide by a monomial like \( -7 m^{2} n^{3}\), you divide each term of the polynomial individually by the monomial.

Take the first term \(7 m^{4} n^{2}\). Dividing by \( -7 m^{2} n^{3}\) gives us \( - m^{2} n^{-1}\), as shown in the exercise. The key here is to perform this division term by term, applying properties of exponents, which we will discuss in the next section, to simplify the expression further. It's important to respect the signs; a positive term divided by a negative monomial will result in a negative term, and vice versa.

To improve your ability to handle these problems, it's crucial to work through several examples and become comfortable with breaking down the polynomial into its individual terms and dividing each one systematically.

Properties of Exponents

When we talk about the properties of exponents, we are referring to the rules that govern how to manipulate powers when multiplying, dividing, or raising them to another power. These properties are the backbone of simplifying expressions with exponents.

For example, when you divide exponents with the same base, you subtract the powers, as seen in the term \(\frac{7 m^{4} n^{2}}{-7 m^{2} n^{3}}\) which simplifies to \( - m^{2-2} n^{2-3}\) or \( - m^{2} n^{-1}\). Another property we apply is that any non-zero number raised to the power of 0 equals 1, as in \(m^{0}\) or \(n^{0}\); this is how the term \( - m n^{0}\) simplifies to \( - m\).

Dealing with negative exponents such as \(n^{-1}\) means that the term is actually the reciprocal of the base raised to the positive exponent: \(n^{-1} = \frac{1}{n}\). Understanding and applying these rules confidently will vastly simplify the process of working with polynomials. Practice applying these properties across various exercises will acquaint students with their application, thus enhancing their problem-solving skills.

Simplifying Expressions

The ultimate goal in algebraic division is to simplify the expression to its most basic form. Simplification makes equations and expressions more manageable and often easier to understand. In the context of dividing polynomials by monomials, once each term of the polynomial is divided by the monomial, we must then combine these results into a single, simplified expression.

In the step-by-step solution provided, after dividing each term, we arrived at \( -\frac{m^{2}}{n} - m + n\). Notice that each term now stands on its own, with no further division possible. Simplification also involves combining like terms whenever possible, but in this case, since there are no like terms, the expression is already in its simplest form.

Encouraging the practice of checking each step can help avoid minor mistakes and retain knowledge on simplification techniques. It's crucial to work through these steps carefully to ensure full comprehension. Restate complex concepts and apply them to real examples, incorporating visual aids such as step-by-step guides or illustrative diagrams, to support the learning process.