Problem 7 - Algebra for College Students

Question

Find each indicated product. Remember the shortcut for multiplying binomials and the other special patterns we discussed in this section. $$-x^{2} y\left(6 x y^{2}+3 x^{2} y^{3}-x^{3} y\right)$$

Step-by-Step Solution

Answer

The product is \(-6x^3y^3 - 3x^4y^4 + x^5y^2\).

Step 1: Distribute the First Term

Multiply \(-x^2 y\) by each term inside the parentheses:1. \(-x^2 y \times 6 x y^2 = -6x^3y^3\)2. \(-x^2 y \times 3 x^2 y^3 = -3x^4y^4\)3. \(-x^2 y \times -x^3 y = x^5 y^2\).Now, combine these results.

Step 2: Combine Results

Write down the results from each multiplication step:\(-6x^3y^3 - 3x^4y^4 + x^5y^2\).

Step 3: Final Simplification

Check the expression for any like terms that can be combined. In this case, there are no like terms, so the simplest form of the expression is already achieved.

Key Concepts

Distributive Property

The distributive property is a key concept in algebra, especially useful when multiplying expressions like polynomials. It allows us to simplify problems by breaking them into more manageable parts. Specifically, when you have an expression like \(a(b + c)\), the distributive property tells us that \(a(b + c) = ab + ac\).

This principle is crucial in polynomial multiplication as well. For instance, in the example \(-x^{2}y (6xy^{2} + 3x^{2}y^{3} - x^{3}y)\), we apply this property by multiplying \(-x^2 y\) with each term inside the parentheses.

• Multiply it by \(6xy^2\), \(3x^2y^3\), and \(-x^3 y\).
• This operation simplifies the expression and makes it easier to handle.

Utilizing the distributive property helps us manage complex polynomial multiplications without getting overwhelmed by the many variables involved.

Binomials

Before diving into multiplication, it’s important to understand what binomials are. A binomial is a type of polynomial that has exactly two terms. These terms are usually separated by a plus or minus sign, for example, \(x + 3\) or \(2a - 5b\).

Binomials are significant because they often simplify complicated algebraic expressions. When multiplying binomials, there are several strategies or shortcuts that can be applied, such as the FOIL method. Though our original exercise involves a monomial multiplying a larger polynomial, understanding how binomials work is foundational.

• Binomials can be present as factors in larger polynomials.
• Mastering binomial multiplication simplifies the understanding of multi-term polynomial multiplication.

Each multiplication of binomials involves applying techniques like the distributive property multiple times, reinforcing the importance of mastering both concepts.

Monomials

Monomials are algebraic expressions that consist of just one term. They can include constants, variables, or both, multiplied together. For example, \(3x\), \(-7y^{2}z\), and \(5\) are all monomials.

When multiplying monomials, things are more straightforward: you multiply the coefficients and then the variables (by adding their exponents if they share a base).

• In our exercise, \(-x^2 y\) is a monomial used to distribute and multiply with a larger polynomial.
• The multiplication involves handling the variables and coefficients systematically to ensure accuracy.

Understanding monomials and how they interact in multiplication is vital to solving complex algebraic expressions. They form the building blocks of larger expressions and mastering them paves the way for proficiency in algebra.