Question

Perform the indicated operation(s) and write the resulting polynomial in standard form.\(-5 y\left(2 y-y^{2}\right)\)

Short answer

\(5y^3 -10y^2\)

Step-by-step solution

Distribute the \(-5y\)

Multiply each term inside the parentheses by \(-5y\), which is known as distribution. So you get \( -5y * 2y\) and \(-5y *-y^2\)

Perform the operations

Now, perform the multiplication operation. \( -5y * 2y = -10y^2 \) and \(-5y*-y^2 = 5y^3\)

Write the Resulting Polynomial in Standard Form

When a polynomial is given in standard form, the exponents decrease from left to right. Hence the answer should be \(5y^3 -10y^2\).

Key concepts

Understanding Distribution in Polynomial Operations

Distribution is a fundamental concept in algebra, especially when dealing with polynomial operations. It revolves around multiplying a single term by each term within a parenthesis. This operation is often called the "Distributive Property."

Here’s how it works: If you have an expression like \(a(b + c)\), you distribute by multiplying \(a\) with both \(b\) and \(c\). The distributive property lets you re-write it as \(ab + ac\). This method is particularly useful for simplifying expressions and solving equations.

In the context of the given exercise, \(-5y(2y - y^2)\),\ the term \(-5y\) is distributed to both \(2y\) and \(-y^2\). It results in two separate products that further simplify the operation, setting the stage for the next steps in polynomial multiplication.

Writing Polynomials in Standard Form

Polynomials are often expressed in what is known as "standard form". The standard form of a polynomial arranges the terms so that their degrees descend from highest exponent to lowest. This arrangement not only brings clarity to the expression but also helps in comparing polynomial sizes.

For example, a polynomial written in standard form looks like this: \ ax^n + bx^{n-1} + cx^{n-2} + \, ... \

After performing operations, it’s crucial to reorder the polynomial to follow this format. Using the exercise example, after distributing and multiplying, you have \(5y^3 - 10y^2\). Since the term with \(y^3\) has the highest degree (3), it comes first, followed by the term with \(y^2\). Emphasizing this form ensures that the polynomial is easily readable and comparable.

Polynomial Multiplication Simplified

Polynomial multiplication involves multiplying terms across two polynomials. This process is similar to distribution but extends to more terms. It's a methodical combination of the distributive property and addition.

Here are a few steps to get started:

• Identify all terms to be multiplied.
• Apply the distributive property systematically to pair each term.
• Combine any like terms for simplification.

In simpler terms, think of it as executing multiple distribution processes. For example, while multiplying \(-5y(2y - y^2)\),\ distribution was used twice: once for each term inside the parentheses. The resulting products, \(-10y^2\) and \(5y^3\), are the results of these multiplications. These results are then combined to form the complete polynomial in standard form.

Mastery of polynomial multiplication requires practice. However, understanding the foundation of distribution makes the process significantly more straightforward. Keep practicing different sets to gain proficiency!