Problem 27 - Algebra A Combined Function

Question

Multiply. \(-y\left(4 x^{3}-7 x^{2} y+x y^{2}+3 y^{3}\right)\)

Step-by-Step Solution

Answer

The result is \(-4yx^3 + 7x^2y^3 - xy^3 - 3y^4\).

Step 1: Understand the Problem

The problem asks us to multiply a single term, \(-y\), by each term in the polynomial \(4x^3 - 7x^2y + xy^2 + 3y^3\).

Step 2: Distribute the First Term

Multiply \(-y\) by \(4x^3\):\[-y \times 4x^3 = -4yx^3\]

Step 3: Distribute the Second Term

Multiply \(-y\) by \(-7x^2y\):\[-y \times (-7x^2y) = 7yx^2y = 7yx^2y^2\]

Step 4: Distribute the Third Term

Multiply \(-y\) by \(xy^2\):\[-y \times xy^2 = -yxy^2 = -yx^1y^2 = -xy^3\]

Step 5: Distribute the Fourth Term

Multiply \(-y\) by \(3y^3\):\[-y \times 3y^3 = -3y^4\]

Step 6: Combine the Terms

Write down the results from the previous steps and combine them into one expression:\[-4yx^3 + 7yx^2y^2 - xy^3 - 3y^4\]This can be simplified to:\[-4yx^3 + 7x^2y^3 - xy^3 - 3y^4\]

Key Concepts

Distributive Property

The distributive property is a fundamental principle in algebra that makes it easier to handle expressions and equations. It allows us to multiply a single term by each term inside a set of parentheses, effectively unlocking the expression to simplify it. In our exercise, we apply the distributive property to a polynomial by multiplying each term inside the polynomial by \(-y\). This means:

• Multiply \(-y\) by each term inside the parentheses.
• Continue this process until every term from the original polynomial is independently multiplied by \(-y\).

This process will transform the polynomial into a collection of terms that are easier to combine or further simplify. Utilizing the distributive property makes complex expressions manageable, breaking them down into simpler, smaller parts.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations. They form the core language of algebra that allows us to represent general mathematical situations. In our exercise, the expression \(4x^3 - 7x^2y + xy^2 + 3y^3\) is an algebraic expression, comprising:

• Variables (\(x\) and \(y\)) which represent unknown or arbitrary numbers.
• Coefficients (numerical factors like 4, -7, 1, and 3) that multiply the variables.
• Operations such as addition and multiplication linking the terms together.

These expressions might look complex, but they're just representations of computations and can be transformed using algebraic principles such as the distributive property. Understanding how to manipulate these expressions enables solving equations and modeling real-world scenarios.

Polynomials

Polynomials are algebraic expressions with one or more terms, where each term contains a variable raised to a non-negative integer power. Our exercise features the polynomial \(4x^3 - 7x^2y + xy^2 + 3y^3\). Here's a breakdown of its components:

• Each term of the polynomial can have multiple variables (e.g., \(x\) and \(y\)) and coefficients.
• The degree of each term is the sum of the exponents of its variables. For instance, \(4x^3\) has a degree of 3, and \(-7x^2y\) has a degree of 3 when considering both variables.
• Polynomials can be classified based on the highest degree of their terms, helping to identify their complexity.

Polynomials are versatile, as they can represent complex functions and equations. By mastering polynomial manipulation, such as multiplication, you can unlock powerful tools for both academic and real-world problem-solving.