Problem 28 - Algebra A Combined Function

Question

Multiply. \(-x\left(6 y^{3}-5 x y^{2}+x^{2} y-5 x^{3}\right)\)

Step-by-Step Solution

Answer

The result is (-6xy^3 + 5x^2y^2 - x^3y + 5x^4).

Step 1: Distribute the (-x) term

To multiply (-x) by the polynomial inside the parentheses, distribute (-x) across each term in the polynomial. This means multiplying (-x) by (6y^3), (-x) by (-5xy^2), (-x) by (x^2y), and (-x) by (-5x^3).

Step 2: Multiply -x by each Term

Calculate each multiplication: 1. (-x) imes 6y^3 = -6xy^3 2. (-x) imes (-5xy^2) = 5x^2y^2 3. (-x) imes x^2y = -x^3y 4. (-x) imes (-5x^3) = 5x^4

Step 3: Write the Resulting Expression

Combine all the terms from the multiplication process to write the final expression:\[-6xy^3 + 5x^2y^2 - x^3y + 5x^4\].

Key Concepts

Distributive Property

The distributive property is a key tool in algebra, especially when dealing with polynomials and other algebraic expressions. It allows you to distribute or "spread" a single term across terms inside parentheses. In simple terms, it means multiplying each term within the parentheses by the outside term. This rule helps break down and simplify expressions, making them easier to manipulate.
For example, if you have a term like \((-x)\) multiplied by a polynomial inside parentheses, you apply the distributive property by multiplying \((-x)\) with each individual term in the polynomial. It's like sharing the \((-x)\) equally with each term inside the parentheses, allowing each term to be multiplied separately.

• For \((-x) \cdot (6y^3)\), you get \(-6xy^3\).
• For \((-x) \cdot (-5xy^2)\), you get \(+5x^2y^2\).
• For \((-x) \cdot (x^2y)\), you get \(-x^3y\).
• For \((-x) \cdot (-5x^3)\), you get \(+5x^4\).

By using the distributive property, we transform a seemingly complex multiplication into simple, manageable steps, making it easier to arrive at the final solution.

Monomials

Monomials are the simplest forms of algebraic expressions. A monomial is essentially a single term that consists of a product of numbers and variables with non-negative integer exponents. It involves multiplication of constants and variables, such as \(3x^2\) or \(-7xy\). Monomials can appear on their own or as part of more complex expressions, like polynomials.
In the exercise, \((-x)\) is a monomial that we distribute across the terms of the polynomial. A key attribute of monomials is that they contain no addition or subtraction, making their structure straightforward.

They have no addition or subtraction in their form.

Monomials are essential building blocks in algebra, forming parts of larger expressions like polynomials.

Polynomials

Polynomials are algebraic expressions that consist of multiple terms, which can include numbers, variables, and exponents. Each term in a polynomial is a monomial, and these terms are typically added or subtracted. Polynomials can vary in complexity, from a simple binomial (two terms) to an extensive polynomial with several terms.
The polynomial in this exercise is what we distribute the monomial \((-x)\) across. The specific polynomial is composed of terms like \(6y^3\), \(-5xy^2\), and so on. Each term in a polynomial adds to its overall degree, determined by the highest power of its variable.

• A polynomial can consist of constants, variables, and exponents.
• Terms are organized in order of degree, with the highest degree term usually written first.
• The degree of a polynomial is based on the term with the highest power of the variable.

The variety of terms gives polynomials a flexibility that is useful in describing a wide range of mathematical situations and problems.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations, forming the building blocks of algebra. These expressions can be as simple as a single variable or as complex as a polynomial containing many terms. Algebraic expressions are used to represent real-world quantities and relationships in a mathematical form.
In our exercise, \(-x(6y^3 - 5xy^2 + x^2y - 5x^3)\), the expression inside the parentheses represents a complex algebraic expression, while \(-x\) is a simpler form.

• Algebraic expressions can involve different operations such as addition, subtraction, multiplication, and division.
• They are used to express mathematical ideas and can be manipulated using properties like associativity, commutativity, and distributivity.
• When dealing with algebraic expressions, especially in multiplying polynomials, understanding the individual parts —such as terms and coefficients — is crucial.

These expressions serve as a crucial tool in everything from solving equations to modeling real-world scenarios, helping bridge the gap between abstract mathematical ideas and practical application.