Question

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

Short answer

\(-18 x^{4} y^{4} - 30 x^{5} y^{6} + 12 x^{4} y^{5}\)

Step-by-step solution

Apply the Distributive Property

Multiply every term inside the parentheses by the term outside, \( -6 x^{3} y^{3} \), which is called the distributive property.

Multiply the Coefficients and Add the Exponents

For each multiplication, multiply the coefficients (numbers) and add the exponents of corresponding variables. Remember that when you multiply powers with the same base, you add their exponents.

Simplify the Expressions

Write the simplified term for each multiplication performed in the previous step, to complete the expansion of the expression.

Key concepts

Polynomial Multiplication

When multiplying polynomials, it's important to apply the distributive property correctly in order to get the resultant polynomial. This includes taking a monomial, such as \(-6 x^{3} y^{3}\), and multiplying it by each term in the polynomial it's being multiplied with, for example \((3 x y + 5 x^{2} y^{3} - 2 x y^{2})\).

Imagine the polynomial you’re multiplying as a set of items you need to distribute evenly. Just as you would give each friend an equal share of your snacks, each term inside the parentheses receives an equal share of the monomial outside. Here's how you would apply this concept systematically: Multiply \(-6 x^{3} y^{3}\) by each term in the parentheses, one at a time, to ensure every term is accounted for. It's like unpacking a box where each item needs to be taken out one at a time and dealt with individually. This ensures that all terms are multiplied accurately, leaving no stone unturned.

Exponents Addition

One of the key rules when dealing with exponentiation in algebra is the rule for multiplying powers with the same base: you add the exponents. For example, when multiplying \(x^{a} \times x^{b}\), the result is \(x^{a+b}\). This rule streamlines the process of multiplying variables in polynomial expressions.

To visualize this, think of exponents as instructions on how many times to use the base as a factor in a multiplication. If you have two sets of these instructions (the exponents), you're essentially combining them when multiplying the bases. This results in a new set of instructions: the sum of the exponents. In our exercise, when multiplying the coefficients of the monomial and polynomial terms, you would add the exponents of \(x\) and \(y\) of like bases. For instance, multiplying \(x^{3}\) by \(x^{2}\) becomes \(x^{3+2}\), simplifying to \(x^{5}\).

Simplifying Expressions

The final step in polynomial multiplication is to simplify the expression you've obtained post multiplication. This involves combining like terms and writing the expression in a more compact form if possible.

Simplifying can be compared to organizing a cluttered room—you group similar items together and arrange them neatly. Here, similar items are like terms, which are terms that have the exact same variable parts regardless of their coefficients. Although our given exercise does not result in like terms to combine, it’s still a crucial step in other cases. After distributing and adding exponents, check if there are any terms that can be combined. This is where you can also adjust signs and coefficients to present the simplest form of the expression. It's about making your final answer as tidy and straightforward as possible, so its true nature is clear and easily understood.