Problem 107 - Intermediate Algebra

Question

Perform each operation. $$ -3 m n^{2}\left(m^{3}-7 m n-2 m^{2}\right) $$

Step-by-Step Solution

Answer

The expanded expression is \(-3m^4n^2 + 21m^2n^3 + 6m^3n^2\).

Step 1: Distribute -3mn^2

To solve the expression \(-3mn^2(m^3 - 7mn - 2m^2)\), we first apply the distributive property across each term inside the parentheses. This requires multiplying each term by \(-3mn^2\).

Step 2: Multiply -3mn^2 with m^3

First, multiply \(-3mn^2\) by \(m^3\). The product is \(-3m^{1+3}n^2\) which simplifies to \(-3m^4n^2\).

Step 3: Multiply -3mn^2 with -7mn

Next, multiply \(-3mn^2\) by \(-7mn\). The calculation is \((-3)(-7)m^{1+1}n^{2+1} = 21m^2n^3\).

Step 4: Multiply -3mn^2 with -2m^2

Then, multiply \(-3mn^2\) by \(-2m^2\). The calculation is \((-3)(-2)m^{1+2}n^2 = 6m^3n^2\).

Step 5: Combine the products

Finally, combine all the distributed terms:\(-3m^4n^2 + 21m^2n^3 + 6m^3n^2\). This gives us the fully expanded expression.

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra. It allows you to break down expressions into more manageable pieces, making it easier to solve complex problems. When you apply the distributive property, you're essentially multiplying a single term by every term within a parenthesis. This can be represented as:

• For any numbers or variables, \(a(b + c) = ab + ac\).

In the given exercise, \(-3mn^2\) was distributed across each term inside the parenthesis \((m^3 - 7mn - 2m^2)\).Each step involved multiplying the term \(-3mn^2\) by each individual component of the polynomial, allowing us to expand and simplify the expression.

Polynomial Expressions

Polynomial expressions consist of variables and constants combined using addition, subtraction, and multiplication. Each part of a polynomial is called a 'term', and terms are usually combined into a sum.

• A polynomial in the variable \(m\) might look like \(3m^2 - 4m + 5\), featuring terms \(3m^2\), \(-4m\), and \(5\).
• The expression within the given exercise, \(m^3 - 7mn - 2m^2\), is a polynomial in two variables: \(m\) and \(n\).

The degree of a polynomial is the highest power of the variable in its terms. Understanding how to handle these terms individually and as part of a larger expression is key to mastering polynomial expressions.

Multiplying Monomials

Multiplying monomials involves multiplying together constants and variables raised to powers. A monomial is a product of constants and variables with non-negative integer exponents.

• For example, when multiplying \((-3mn^2)\) with another monomial \(m^3\), the approach is to multiply the coefficients (numbers in front of variables) and then apply the product rule for exponents.
• The product rule states, for any numbers \(a\) and \(b\), and any variables \(x\), \(x^a \cdot x^b = x^{a+b}\).

Thus, when multiplying \(-3mn^2\) by \(m^3\), you combine like bases: \(m^{1+3}\) and \(n^{2+0}\).The result is \(-3m^4n^2\).Understanding this principle is crucial for simplifying products of expressions efficiently.