Problem 74 - Elementary Algebra

Question

\((\) Section 4.6\()\) Find the product. \((a+4)\left(a^{2}-2 a+3\right)\)

Step-by-Step Solution

Answer

Question: Find the product of the expressions $(a+4)$ and $(a^2-2a+3)$. Answer: $a^3 + 2a^2 - 5a + 12$

Step 1: Distribute the first term in the first expression to all terms in the second expression

We will first distribute the term \(a\) in expression \((a+4)\) to every term in the second expression \((a^2-2a+3)\). This will result in: $$ a(a^2) + a(-2a) + a(3) $$

Step 2: Distribute the second term in the first expression to all terms in the second expression

Next, we will distribute the term \(4\) in the expression \((a+4)\) to every term in the second expression \((a^2-2a+3)\). This will result in: $$ 4(a^2) + 4(-2a) + 4(3) $$

Step 3: Combine the distributed terms

We will now combine the terms we got from Step 1 and Step 2: $$ (a(a^2) + a(-2a) + a(3)) + (4(a^2) + 4(-2a) + 4(3)) $$

Step 4: Simplify the equation

Now, we will simplify the equation by multiplying the terms and combining the like terms. This means we add the coefficients of the terms with the same powers of \(a\). The final expression will be: $$ (a^3 - 2a^2 + 3a) + (4a^2 - 8a + 12) $$ Combine the like terms: $$ a^3 + (-2a^2 + 4a^2) +( 3a - 8a) + 12 $$ $$ a^3 + 2a^2 - 5a + 12 $$ So the product of \((a+4)\) and \((a^2-2a+3)\) is: $$ a^3 + 2a^2 - 5a + 12 $$

Key Concepts

Distribution Property

The distribution property is a fundamental principle in algebra applied when multiplying one term by a group of terms inside a parenthesis. In the context of polynomial multiplication, this property helps us simplify expressions by expanding terms. Let's illustrate this with the original problem:Given the expression

• \((a+4)(a^2-2a+3)\)

we use distribution to spread each term of the binomial \((a+4)\) across every term in the trinomial \((a^2-2a+3)\). This involves two separate distributions:- First, multiply \(a\) (the first term of \((a+4)\)) with all parts of \((a^2-2a+3)\), giving us: - \(a \cdot a^2 = a^3\) - \(a \cdot (-2a) = -2a^2\) - \(a \cdot 3 = 3a\)
- Second, multiply \(4\) (the second term of \((a+4)\)) with all parts of \((a^2-2a+3)\), resulting in: - \(4 \cdot a^2 = 4a^2\) - \(4 \cdot (-2a) = -8a\) - \(4 \cdot 3 = 12\)The distribution property allows us to break down complex problems into manageable steps, making it a crucial tool in understanding polynomial multiplication.

Combining Like Terms

After applying the distribution property, you are left with multiple terms that need to be simplified. **Combining like terms** takes the scattered terms and merges them to make the expression simpler and more readable. Like terms in algebra are terms that have the same variable raised to the same power. For example:- In the expression: - \(a^3 - 2a^2 + 3a + 4a^2 - 8a + 12\) - you combine like terms based on their degree: - Combine \(-2a^2\) and \(4a^2\), resulting in \(2a^2\) - Combine \(3a\) and \(-8a\), resulting in \(-5a\)So, simplifying, you combine and retain:- \(a^3 + 2a^2 - 5a + 12\)Using this approach ensures your polynomial expression is coherent and simplified for further use or evaluation. This step is essential in polynomials, as it reduces complexity and helps identify the polynomial structure more readily.

Algebraic Expressions

Algebraic expressions are representations of numbers and operations expressed through variables and constants. In our exercise, expressions like

• \((a+4)\)
• \((a^2-2a+3)\)

form the backbone of polynomial mathematics. Here, variables represent unknown values, while constants provide definite values.**Polynomial Expression**- A polynomial is a type of algebraic expression that consists of variables and coefficients, created by adding, subtracting, and multiplying terms together.- Polynomial multiplication includes terms like: - \(a^3\) (third degree) - \(2a^2\) (second degree) - \(-5a\) (first degree) - \(+12\) (zero degree or constant)Through operations like distribution and combining like terms, we harness the power of algebraic expressions to solve, simplify, or rearrange complex equations effectively. Understanding how to manipulate these expressions efficiently is key to mastering algebra and tackling a variety of mathematical challenges.