Problem 9 - Algebra for College Students

Question

Find each product. $$(5 x y)\left(-6 y^{3}\right)$$

Step-by-Step Solution

Answer

The product is \(-30xy^4\).

Step 1: Identify Terms Involved

The expression given is \((5xy)(-6y^3)\). Here, \(5xy\) and \(-6y^3\) are the two terms that need to be multiplied.

Step 2: Multiply Coefficients

Both terms have coefficients (numeric parts). Multiply the coefficients: \(5\) and \(-6\). This gives us \(5 \times (-6) = -30\).

Step 3: Multiply Variable \(x\)

The first term has a variable \(x\), while the second term does not. So, \(x\) remains as is in the product.

Step 4: Multiply Variable \(y\) Parts

The variables \(y\) in both terms are combined: \(y\) from the first term and \(y^3\) from the second term. Using the property \(y^a \times y^b = y^{a+b}\), we get \(y^{1+3} = y^4\).

Step 5: Combine Results

Combine all parts. The final product is \(-30xy^4\).

Key Concepts

Coefficients Multiplication

Understanding coefficients is crucial when learning about polynomial multiplication. Coefficients are the numerical parts of terms in a polynomial. In our exercise, the terms are \(5xy\) and \(-6y^3\). The coefficients are \(5\) and \(-6\), respectively.

To multiply these coefficients, simply multiply the numbers as you normally would in arithmetic. Here, you multiply \(5\) and \(-6\) to get \(-30\).

• Step 1: Take the coefficients of each term, \(5\) and \(-6\).
• Step 2: Multiply them. Since \(5 \times (-6)\) equals \(-30\), this becomes the new coefficient of the product.

This multiplication of coefficients is the foundational step before you move on to handling the variable parts of polynomial multiplication.

Exponent Rules

Exponent rules are key when multiplying variables in algebraic expressions. They make the process of working with powers straightforward and manageable. A common situation is multiplying like bases with exponents.

In our exercise, we encounter the term \(y\) from \(5xy\) and \(y^3\) from \(-6y^3\). These share the base \(y\). To multiply them, you add their exponents according to the rule \(y^a \times y^b = y^{a+b}\). It's important to remember to check both terms for exponents, even if it isn't written. The base \(y\) in \(5xy\) is technically \(y^1\).

• Step 1: Identify the exponents. In \(5xy\), the \(y\) is \(y^1\), and in \(-6y^3\), the exponent is \(3\).
• Step 2: Add the exponents: \(1 + 3 = 4\).
• Step 3: Write the result as \(y^4\).

Applying exponent rules simplifies multiplication and solves the expression effectively without any errors.

Variable Multiplication

When multiplying polynomials, handling variables correctly is essential. Not only do you focus on the coefficients, but you must also carefully multiply all variables involved in each term.

Consider the expression \((5xy)(-6y^3)\):

• Variables in the first term: \(x\) and \(y\). In the second term: \(y^3\).
• To multiply variables, multiply like variables together using exponent rules and keep different ones intact.

In this product, only\(y\) is common. Apply the exponent rule to get \(y^4\). The variable \(x\) doesn't have a counterpart in \(-6y^3\) and remains as is. The variables combine to perform the multiplication:
\[(5xy) \times (-6y^3) = -30xy^4\]
In summary, identify every variable from each term, multiply like variables, and transfer any solitary variables directly into your final expression. This approach ensures accuracy and clarity in polynomial multiplication.