Problem 48 - Algebra A Combined Function

Question

Multiply. \((a+2)\left(a^{3}-3 a^{2}+7\right)\)

Step-by-Step Solution

Answer

The product is \(a^4 - a^3 - 6a^2 + 7a + 14\).

Step 1: Apply the Distributive Property

To multiply \(a+2\) by \(a^3 - 3a^2 + 7\), we use the distributive property. This involves multiplying each term in \(a+2\) by each term in the polynomial \(a^3 - 3a^2 + 7\).

Step 2: Distribute First Term (a)

First, distribute the \(a\) in \(a+2\) across each term of the polynomial: \[ a imes a^3 = a^4, \ a imes (-3a^2) = -3a^3, \ a imes 7 = 7a \] This gives us: \[ a^4 - 3a^3 + 7a \]

Step 3: Distribute Second Term (2)

Now, distribute the \(2\) in \(a+2\) across each term of the polynomial: \[ 2 imes a^3 = 2a^3, \ 2 imes (-3a^2) = -6a^2, \ 2 imes 7 = 14 \] This gives us: \[ 2a^3 - 6a^2 + 14 \]

Step 4: Combine Like Terms

Combine the results from Step 2 and Step 3: \[ (a^4 - 3a^3 + 7a) + (2a^3 - 6a^2 + 14) \] Combine like terms to simplify:\[ a^4 + (-3a^3 + 2a^3) - 6a^2 + 7a + 14 \] This simplifies to: \[ a^4 - a^3 - 6a^2 + 7a + 14 \]

Step 5: Final Expression

The final result of multiplying \(a+2\) by \(a^3 - 3a^2 + 7\) is: \[ a^4 - a^3 - 6a^2 + 7a + 14 \]

Key Concepts

Distributive Property of Multiplication

When multiplying algebraic expressions, the distributive property is a handy tool. It allows you to break down complex expressions into simpler pieces. The rule is simple: each term in one expression must be multiplied by each term in the other expression. This step-by-step approach ensures that no component is left behind.

In our exercise, we used the distributive property to multiply \((a+2)\) by \(a^3 - 3a^2 + 7\). Here's how it works:

• Multiply each term inside the parenthesis \((a+2)\) by every term in the other polynomial \(a^3 - 3a^2 + 7\).
• First, distribute \(a\) to each term: \(a \times a^3, a \times (-3a^2), a \times 7\).
• Next, distribute \(2\) to every term: \(2 \times a^3, 2 \times (-3a^2), 2 \times 7\).

Thus, we methodically apply the distributive property to get multiple terms that will later be combined into a single solution. It is a crucial algebraic tool that simplifies the multiplication of polynomials, ensuring all parts of the expressions are included in the final answer.

Combining Like Terms

Combining like terms is a fundamental concept for simplifying algebraic expressions. After using the distributive property, the resulting expression may have several similar terms. We simplify the polynomial by adding or subtracting these similar terms.

• Like terms are terms that have the same variable raised to the exact power.
• In our exercise, once we distributed all terms, we had \(a^4, -3a^3, 7a\) from one set and \(2a^3, -6a^2, 14\) from another.
• Combine \(-3a^3\) and \(2a^3\), as these both are terms with \(a^3\), to get \(-a^3\).

This reduction process is essential because it reorganizes and simplifies expressions by combining similar components, making them clearer and more manageable.

Understanding Algebraic Expressions

An algebraic expression is a mathematical phrase that includes numbers, variables, and operation symbols. Understanding these expressions is fundamental for manipulating and solving equations.

• Variables, like \(a\), serve as placeholders that represent numbers. These can vary and are typically used to generalize math problems.
• Expressions can be added, subtracted, multiplied, and divided, as we have done in this exercise.
• Our exercise started with the expression \((a+2)(a^3 - 3a^2 + 7)\), a typical algebraic expression formed by two multiplied polynomials.

Knowing how to work with algebraic expressions allows you to solve problems systematically. Each part fits into mathematical manipulation processes that lead to finding solutions. Remember, practice is key in mastering handling these expressions effectively.