Problem 50 - Algebra 1: Concepts and Skills

Question

$$ (a-3)\left(a^{2}-4 a-6\right) $$

Step-by-Step Solution

Answer

The expanded form of \((a-3)\left(a^{2}-4 a-6\right)\) is \(a^{3} - 7a^{2} + 6a + 18\)

Step 1: Distribute (a-3) to each term in the second bracket

To begin, multiply \(a-3\) by each term in the bracket \(\left(a^{2}-4 a-6\right)\). This results in \(a(a^{2}) - a(4a) - a(6) - 3(a^{2}) + 3(4a) + 3(6)\)

Step 2: Simplify individual terms

Next, simplify each multiplication operation from step 1. This results in \(a^{3} - 4a^{2} - 6a - 3a^{2} + 12a + 18\)

Step 3: Combine like terms

Finally, add up all the terms with the same power of \(a\). This results in \(a^{3} - 7a^{2} + 6a + 18\)

Key Concepts

Distributive Property

The distributive property is a cornerstone of algebra that allows students to multiply a single term by a set of terms in parentheses. The rule states that when you have a multiplication of a term by a group of terms added or subtracted together, you need to 'distribute' the multiplication to each term inside the parentheses. For instance, in the expression
\((a-3)(a^2-4a-6)\),
we distribute the \(a-3\) across the terms within the parentheses \(a^2\), \(-4a\), and \(-6\), one by one. This means we perform three individual multiplication operations: \(a\) times \(a^2\), \(a\) times \(-4a\), and \(a\) times \(-6\), and similarly, \(-3\) with each term. Distributive property promotes understanding that multiplication is a binary operation affecting each term in the second binomial individually, leading to a correct simplification of complex expressions. This is essential because it helps students avoid common mistakes, such as combining terms that should not be combined or misapplying the operation to only part of the expression.

Simplifying Expressions

Once the distributive property has been applied, the next logical step is simplifying the expression. Simplifying involves executing the arithmetic operations resulting from the distribution and reducing the expression to its simplest form. To simplify the terms from our distribution, we carry out the multiplication, converting expressions like \(a(a^2)\), \(a(-4a)\), and \(a(-6)\) into \(a^3\), \(-4a^2\), and \(-6a\) respectively.
The process is repeated with \(-3\) distributed across the terms in the parentheses, resulting in \(-3a^2\), \(3(4a)\), and \(3(-6)\).

Simplifying Step-by-Step

• Multiply coefficients with variables
• Apply exponents
• Perform any addition or subtraction

This stage is crucial for making the further steps like combining like terms less confusing. Clear simplification paves the way for easily identifying like terms, which is a vital aspect of working with polynomials.

Combining Like Terms

Once you have simplified the expression, your next task in polynomial multiplication, as in our example, is to combine like terms. Like terms are terms in an expression that have the exact same variables raised to the exact same power. For example, in our simplified expression, \(a^3 - 4a^2 - 6a - 3a^2 + 12a + 18\), the like terms are \(-4a^2\) and \(-3a^2\), as well as \(-6a\) and \(12a\). To combine these, we add or subtract them based on their coefficients, resulting in \(-7a^2\) and \(6a\), respectively.
The final expression includes only distinct powers of \(a\) and any constants:
\(a^3 - 7a^2 + 6a + 18\).

Tips for Combining Like Terms

• Identify terms with the same variable and exponent
• Add or subtract the coefficients
• Write the resultant terms without altering the variables or exponents

In combining like terms, it is important to keep the variables and their exponents intact to maintain the integrity of the polynomial. Incorrectly combining unlike terms will lead to incorrect solutions and misunderstandings in algebraic concepts.