Problem 100 - Intermediate Algebra

Question

Simplify each expression. $$9\left(m^{3}+3\right)-5\left(3-m^{3}\right)-8\left(-1-m^{3}\right)$$

Step-by-Step Solution

Answer

The simplified expression is \(22m^3 + 20\).

Step 1: Expand each group

First, we need to distribute the constants into each expression inside the parentheses.For the term \(9(m^3 + 3)\), distribute 9:\[9(m^3 + 3) = 9m^3 + 27\]For the term \(-5(3 - m^3)\), distribute -5:\[-5(3 - m^3) = -15 + 5m^3\]For the term \(-8(-1 - m^3)\), distribute -8:\[-8(-1 - m^3) = 8 + 8m^3\]

Step 2: Combine like terms

Now, combine all the expanded terms together:\[9m^3 + 27 - 15 + 5m^3 + 8 + 8m^3\]Next, combine the like terms. First, combine all terms with \(m^3\):\[9m^3 + 5m^3 + 8m^3 = 22m^3\]Then, combine the constant terms:\[27 - 15 + 8 = 20\]The expression now becomes:\[22m^3 + 20\]

Step 3: Final expression

Once all like terms are combined, verify the expression. The simplified expression is:\[22m^3 + 20\]

Key Concepts

Polynomial Expansion

Polynomial expansion is a crucial step when dealing with expressions like the one in our exercise. Think of it as breaking down the terms inside parentheses and then spreading, or distributing, the outer number to each term within the parentheses. This allows us to simplify the expression by considering each term separately. For example, in the term \(9(m^3 + 3)\), the number 9 needs to be multiplied by each term inside the parentheses individually. Breaking it down, you multiply 9 by \(m^3\) to obtain \(9m^3\), and then multiply 9 by 3 to get 27. Thus, \(9(m^3 + 3)\) becomes \(9m^3 + 27\).
Another example in this problem is with the term \(-5(3 - m^3)\). Multiply \(-5\) by 3 to get \(-15\), and \(-5\) by \(-m^3\) to get \(5m^3\). This results in \(-15 + 5m^3\). Polynomial expansion helps set the stage for further simplification by making terms easier to manage individually.

Distributive Property

The distributive property is a fundamental rule in algebra that facilitates polynomial expansion. This property states that for any three numbers \(a, b,\) and \(c\), the property is used in the form \(a(b + c) = ab + ac\). It's like distributing the "a" across each term in the parentheses "b" and "c".
Let's apply this to our problem once more using \(-8(-1-m^3)\). We distribute \(-8\) by multiplying it with \(-1\) and with \(-m^3\). So, \(-8\) multiplied by \(-1\) becomes 8, because two negatives make a positive. Then, \(-8\) multiplied by \(-m^3\) becomes \(8m^3\), resulting in \(8 + 8m^3\).

• Distributive property simplifies expressions by removing parentheses.
• This property is essential for solving expressions with multiple terms needing expansion.

After applying the distributive property correctly, moving on to combining like terms becomes much easier.

Combining Like Terms

Combining like terms is the simplifying step where terms that have the same variable and exponent are added together, and constant numbers are combined similarly. The objective here is to bring expressions to their simplest form.
For instance, in our example, after polynomial expansions, we had the terms \(9m^3, 5m^3,\) and \(8m^3\). These are 'like terms' because they all contain \(m^3\) as their variable component. So, simply add their coefficients (9 + 5 + 8) which results in \(22m^3\).
Similarly, the constant terms 27, -15, and 8 need to be combined. Adding these gives us 20 (27 - 15 + 8). Thus, by combining like terms, the expanded expression \(9m^3 + 27 - 15 + 5m^3 + 8 + 8m^3\) is simplified to \(22m^3 + 20\).

• Like terms have identical variables and exponents.
• Combining these terms aids in simplifying the expression to its shortest form.

Understanding how to combine like terms can make complex algebraic expressions easier to work with and solve.