Problem 25 - Elementary Algebra

Question

For the following problems, perform the divisions. $$ \frac{7 x^{3} y+8 x^{2} y^{3}+3 x y^{4}-4 x y}{x y} $$

Step-by-Step Solution

Answer

Question: Perform the division of the given polynomial by the monomial: \((7x^3y + 8x^2y^3 + 3xy^4 - 4xy) \div (xy)\). Answer: When divided by \((xy)\), the given polynomial is equal to \(7x^2 + 8xy^2 + 3y^3 - 4\).

Step 1: Divide each term of the polynomial by the monomial

To perform the division, divide each term of the given polynomial by the monomial \((xy)\).

Step 2: Divide the first term

Divide the first term \((7x^3y)\) by \((xy)\): $$ \frac{7x^3y}{xy} = 7x^2 $$

Step 3: Divide the second term

Divide the second term \((8x^2y^3)\) by \((xy)\): $$ \frac{8x^2y^3}{xy} = 8xy^2 $$

Step 4: Divide the third term

Divide the third term \((3xy^4)\) by \((xy)\): $$ \frac{3xy^4}{xy} = 3y^3 $$

Step 5: Divide the fourth term

Divide the fourth term \((-4xy)\) by \((xy)\): $$ \frac{-4xy}{xy} = -4 $$

Step 6: Write down the result

Combine the results from steps 2 to 5 to get the final answer: $$ 7x^2 + 8xy^2 + 3y^3 - 4 $$

Key Concepts

Monomials

Monomials are the building blocks of polynomials. They consist of a single term, which can be a constant, a variable, or a product of constants and variables. Each variable in a monomial can have non-negative integer exponents. For example, in the expression \(7x^3y\),

• 7 is the coefficient
• \(x^3\) and \(y\) are the variables with exponents

Monomials are simple, yet they can be powerful when combined to form polynomials. When dividing polynomials by a monomial, it's important to simplify each term separately by the monomial, as shown in the given exercise. Identifying and working with monomials correctly is crucial as it helps maintain the structure and meaning of the expression during operations.

Exponents

Exponents are numerical symbols that denote the number of times a base is multiplied by itself. In the context of polynomial expressions, exponents play a vital role in determining the degree and complexity of the polynomial. For example, in the term \(x^3\), 3 is the exponent that indicates \(x\) is multiplied by itself three times. When dividing terms with exponents, you subtract the exponent of the divisor from the exponent of the dividend. Thus, when dividing \(7x^3y\) by \(xy\), you subtract the exponents of \(x\) and \(y\) from \(x^3\) and \(y\) respectively:

• \(x^{3-1} = x^2\)
• \(y^{1-1} = y^0 = 1\)

Understanding how to manipulate exponents ensures that division and other operations are performed accurately. It is also vital for simplifying terms, reducing them to their simplest form.

Simplifying Expressions

Simplifying expressions involves rewriting them in a way that is easier to interpret and solve. This often means reducing them to their simplest terms by performing basic algebraic operations such as combining like terms, factoring, and expanding as needed. In polynomial division, as shown in the example, simplifying involves dividing each term by the given monomial and reducing them. Here's how to simplify:

• Identify and perform operations on similar terms.
• Apply properties of exponents, such as power reduction by subtraction during division.
• Combine and simplify results to ensure the expression is as straightforward as possible.

Effective simplification reveals a clearer understanding of the polynomial structure and sometimes unveils inherent solutions or simplifications that were not evident in the unsimplified form. By breaking down complex expressions into simpler forms, students can tackle more intricate mathematics with ease.