Problem 105 - Intermediate Algebra

Question

Perform each operation. $$ \left(a^{2}-4 a-3\right)(a-2) $$

Step-by-Step Solution

Answer

The result of the operation is: \(a^3 - 6a^2 + 5a + 6\).

Step 1: Understand the Problem

We need to perform the operation of multiplying the polynomial \(a^2 - 4a - 3\) by \(a - 2\). This involves distributing each term in the second polynomial to every term in the first polynomial.

Step 2: Distribute the First Term

Multiply \(a\) from \(a - 2\) by each term in \(a^2 - 4a - 3\). \[ a imes (a^2 - 4a - 3) = a(a^2) - a(4a) - a(3) \]Simplifying this, we get:\[ a^3 - 4a^2 - 3a \]

Step 3: Distribute the Second Term

Multiply \(-2\) from \(a - 2\) by each term in \(a^2 - 4a - 3\).\[ -2 imes (a^2 - 4a - 3) = -2(a^2) + 2(4a) + 2(3) \]Simplifying this, we get:\[ -2a^2 + 8a + 6 \]

Step 4: Combine Like Terms

Now, add the results of the two distributions together:\[ a^3 - 4a^2 - 3a - 2a^2 + 8a + 6 \]Combine the like terms:- The \(a^2\) terms: \(-4a^2 - 2a^2 = -6a^2\)- The \(a\) terms: \(-3a + 8a = 5a\)The combination gives the final expression:\[ a^3 - 6a^2 + 5a + 6 \]

Step 5: Final Solution

The product of the given polynomials \((a^2 - 4a - 3)(a - 2)\) is:\[ a^3 - 6a^2 + 5a + 6 \]

Key Concepts

Distribution in Algebra

Distribution in algebra is a powerful tool, especially when it comes to multiplying polynomials. When you see a problem like \((a^2 - 4a - 3)(a - 2)\), you use the distributive property. This means that each term in the second polynomial, \(a - 2\), is multiplied by every term in the first polynomial, \(a^2 - 4a - 3\).

Start by multiplying each term of \(a - 2\) separately:

• Multiply \(a\) by each term in \(a^2 - 4a - 3\). This gives \(a(a^2) - a(4a) - a(3)\), simplifying to \(a^3 - 4a^2 - 3a\).
• Then, multiply \(-2\) by each of the same terms: \(-2(a^2) + (-2)(-4a) + (-2)(-3)\), resulting in \(-2a^2 + 8a + 6\).

The distributive property helps ensure every term gets properly multiplied, setting up the problem for the next critical step: combining like terms.

Combining Like Terms

After distributing each term from one polynomial to another, you often end up with terms that can be combined to simplify the expression. This process is called combining like terms. In our example, after distributing, we get:\[ a^3 - 4a^2 - 3a - 2a^2 + 8a + 6 \]

Now, look at the terms closely:

• The \(a^2\) terms: \(-4a^2\) and \(-2a^2\), which combine to \(-6a^2\).
• The \(a\) terms: \(-3a\) and \(8a\), which combine to \(5a\).
• Finally, the other terms (\(a^3\) and the constant \(6\)) stand alone, as there are no other terms like them in the expression.

Combining like terms simplifies the polynomial to:\[ a^3 - 6a^2 + 5a + 6 \]
This simplification step is crucial as it reduces the polynomial to its most compact and simplified form.

Polynomial Operations

Polynomial operations include addition, subtraction, and multiplication of polynomial expressions. In this exercise, we focused on multiplication, which involves using the distributive property effectively.

Whenever you multiply polynomials, like in \((a^2 - 4a - 3)(a - 2)\), it's important to be systematic:

• Use distribution to multiply each term in one polynomial by every term in the other polynomial.
• After multiplication, combine like terms for simplification.

To solve polynomial multiplication problems efficiently:

1. Understand the structure of each polynomial (e.g., how many terms it has, the degree of each term).
2. Perform the distributive steps methodically.
3. Finally, ensure all like terms are combined correctly for a simplified solution.

These polynomial operations are fundamental in algebra, providing the groundwork for more complex mathematical concepts and calculations. Mastery of these techniques facilitates handling polynomials effectively in various mathematical contexts.