Question

$$6 m n^{2}\left(5 m^{3} n+4 m n^{2}-2 m^{2} n+m n\right)$$

Short answer

\[30 m^4 n^3 + 24 m^2 n^4 - 12 m^3 n^3 + 6 m^2 n^3\]

Step-by-step solution

Distribute the term outside the parentheses

Apply the distributive property \(a(b+c) = ab + ac\) to multiply \(6 m n^{2}\) with each term inside the parentheses.

Multiply the constants and variables separately

Multiply the numerical coefficients and then use the power rule \(a^{m}a^{n} = a^{(m+n)})\) for multiplying the variables with the same base.

Combine like terms if necessary

After the distribution, check if there are any like terms. In this case, there are no like terms to combine, so we move on to the final answer.

Key concepts

Power Rule for Exponents

Understanding the power rule for exponents is crucial when dealing with algebraic expressions. The power rule states that when you multiply two powers with the same base, you simply add the exponents:
\( a^m \times a^n = a^{(m+n)} \)
For example, if you have \( m^2 \times m^3 \), the base here is \( m \), and the exponents are 2 and 3. Applying the power rule, you get \( m^{2+3} = m^5 \). This comes in handy, particularly when multiplying polynomials, as you often encounter variables raised to different powers. It's essential not to confuse this rule with the rules for adding or subtracting exponents, which are different mathematical operations. The power rule is a simple yet powerful tool that enables you to simplify expressions quickly and manage complex polynomial equations with ease.

Multiplying Polynomials

Multiplying polynomials is a critical skill in algebra that involves combining two or more polynomials by distributing each term of the first polynomial to every term of the second. This process, commonly known as FOIL (First, Outer, Inner, Last) when dealing with binomials, can be extended to any polynomials.

• First, distribute each term of the multiplier polynomial to each term of the multiplicand polynomial.
• Second, apply the power rule for exponents when terms of the same base are multiplied.
• Finally, always remember to multiply the coefficients (numerical factors) as well.

For instance, if we take \(6 m n^{2}\) and multiply it by each term in \(5 m^{3} n + 4 m n^{2} - 2 m^{2} n + m n\), we should distribute the \(6 m n^{2}\) across the polynomial inside the parenthesis, ensuring to multiply both the coefficients and variables separately as per Step 2 in the solutions.

Combining Like Terms

Once you have finished multiplying polynomials, the next step often involves combining like terms. Like terms are terms in an expression that have the same variables raised to the same powers, though they can have different coefficients. To combine like terms, one simply adds or subtracts the coefficients and keeps the variable part unchanged.

Take the following for example, after distributing and multiplying, you might end up with terms like \(12 m^2 n^2\) and \(3 m^2 n^2\). Since they have the identical variables raised to the same powers, they can be combined to \(15 m^2 n^2\).

This simplification step is pivotal for achieving the most reduced form of an algebraic expression. Even if the exercise initially doesn't appear to contain like terms, always review your final expression; combining like terms could still be necessary after multiplication has altered the exponents on your variables.