Question

Multiply the polynomials using the FOIL method. Express your answer as a single polynomial in standard form. $$ (-2 x-3 y)(3 x+2 y) $$

Short answer

-6x^2 - 13xy - 6y^2

Step-by-step solution

Distribute the First Terms

Multiply the first terms in each parenthesis. This is \(-2x \times 3x = -6x^2\).

Distribute the Outer Terms

Multiply the outer terms. This is \(-2x \times 2y = -4xy\).

Distribute the Inner Terms

Multiply the inner terms. This is \(-3y \times 3x = -9xy\).

Distribute the Last Terms

Multiply the last terms. This is \(-3y \times 2y = -6y^2\).

Combine Like Terms

Combine the middle terms to simplify \(-4xy - 9xy = -13xy\). Then write the final polynomial: \(-6x^2 - 13xy - 6y^2\).

Key concepts

Polynomial Multiplication

Polynomial multiplication is the process of multiplying two polynomials together to produce a single polynomial. This involves distributing each term of the first polynomial to every term of the second polynomial. Understanding polynomial multiplication is essential because it is a fundamental algebraic process found in many areas of math. In our example, we multiply the polynomials \( (-2x-3y)(3x+2y) \). The steps involve distributing each term carefully, ensuring that all combinations of the terms are considered.

Distributive Property

The distributive property is a key algebraic property that helps in polynomial multiplication. It states that for any three numbers a, b, and c, the expression a(b + c) is equivalent to ab + ac. This principle allows us to distribute terms across parentheses when multiplying polynomials. For example, in our exercise: \( (-2x - 3y)(3x + 2y) \), we use the distributive property to expand each term as follows:

\t
• First, \( -2x \) is multiplied by both \( 3x \) and \( 2y \).
\t
• Next, \( -3y \) is multiplied by both \( 3x \) and \( 2y \).

This distribution forms the basis for the FOIL method, helping us keep track of all multiplications needed.

Like Terms

Combining like terms is the final step in simplifying a polynomial expression. Like terms are terms that have the same variables raised to the same power. After distributing all terms using the FOIL method, you often end up with several terms that can be combined. For example, in our example, the multiplication results in: \( -6x^2 - 4xy - 9xy - 6y^2 \). Here, \( -4xy \) and \( -9xy \) are like terms because both have the variables \( xy \). Combining them gives us \( -13xy \). The final simplified polynomial is \( -6x^2 - 13xy - 6y^2 \).