Problem 17 - Beginning and Intermediate Algebra

Question

Multiply. $$2 x^{3} y\left(x y^{2}+8 x y-11 y+2\right)$$

Step-by-Step Solution

Answer

The short answer to the question is: \(2x^4y^3 + 16x^4y^2 - 22x^3y^2 + 4x^3y\)

Step 1: Identify terms of the polynomial and monomial

The monomial is \(2x^3y\), and the polynomial consists of four terms: \(xy^2\), \(8xy\), \(-11y\), and \(2\).

Step 2: Multiply monomial with each term in the polynomial

Now, we'll multiply the monomial with each term in the polynomial: 1. \(2x^3y \cdot xy^2 = 2x^4y^3\) 2. \(2x^3y \cdot 8xy = 16x^4y^2\) 3. \(2x^3y \cdot -11y = -22x^3y^2\) 4. \(2x^3y \cdot 2 = 4x^3y\)

Step 3: Combine terms

Finally, we need to combine the four products from Step 2 to obtain the resulting expression: \(2x^4y^3 + 16x^4y^2 - 22x^3y^2 + 4x^3y\)

Key Concepts

Monomials

A monomial is an algebraic expression that consists of a single term. This term can be a constant, a variable, or a product of constants and variables raised to a power. Monomials do not include addition or subtraction, which distinguishes them from more complex expressions. For example, in the expression given in the exercise, the monomial is \(2x^3y\). This monomial includes:

• A constant: 2
• Variables: \(x\) and \(y\)
• Exponents: \(x^3\) indicates that \(x\) is multiplied by itself three times

When multiplying a monomial by another expression, each component (constant, variable, and exponent) interacts with the terms of the expression it's being multiplied by. Understanding monomials is crucial as they form the basic building block for more complex expressions like polynomials.

Polynomials

Polynomials are a more complex type of algebraic expression made up from the sum of monomials. Each monomial within a polynomial is referred to as a "term" of the polynomial. In the exercise, the polynomial is \(xy^2 + 8xy - 11y + 2\), consisting of four different terms. Here's a breakdown of these terms:

• \(xy^2\) is a product of \(x\) and \(y^2\)
• \(8xy\) includes a constant \(8\) multiplied with \(xy\)
• \(-11y\) combines the constant \(-11\) and the variable \(y\)
• \(2\) is a constant

Polynomials can have different degrees depending on the highest power of the variables present in the expression. In operations involving polynomials, it's important to handle each term correctly, especially when multiplying with a monomial, as it affects the resulting expression significantly.

Algebraic Expressions

Algebraic expressions are combinations of variables, constants, and arithmetic operations which follow the rules of algebra. They can range from simple expressions like monomials to more complex structures like polynomials and beyond. An algebraic expression can represent numbers, quantities, or both. In the exercise, the entire expression \(2x^3y(xy^2 + 8xy - 11y + 2)\) is an example of an algebraic expression.When dealing with algebraic expressions, it's important to understand how to simplify them. Simplification often involves combining like terms, factoring, or expanding the expression. In polynomial multiplication, as demonstrated in the exercise, the expression is simplified by distributing the monomial across each term of the polynomial. This step-by-step approach helps ensure that each part of the expression has been accounted for in the final solution, keeping the expression organized and manageable.