Problem 69 - Elementary Algebra

Question

For the following problems, simplify each of the algebraic expressions. $$ A(A+7)+4\left(A^{2}+3 a+1\right) $$

Step-by-Step Solution

Answer

Question: Simplify the algebraic expression $A(A+7) + 4(A^{2} + 3A + 1)$. Answer: The simplified expression is $5A^{2} + 19A + 4$.

Step 1: Distribute terms

We first distribute the terms within each parenthesis and the brackets, which means multiplying the term outside the parenthesis with each term inside. $$ A(A+7) + 4(A^{2} + 3A + 1) $$

Step 2: Expand the expressions

Now, we will multiply through the terms as indicated. $$ A^{2} + 7A + 4A^{2} + 12A + 4 $$

Step 3: Combine like terms

Combine the terms, which have the same variables and exponents. $$ (A^{2} + 4A^{2}) + (7A + 12A) + 4 $$

Step 4: Simplify the resulting expression

Add the coefficients of the like terms to simplify. $$ 5A^{2} + 19A + 4 $$ So, the simplified expression for the given problem is: $$ 5A^{2} + 19A + 4 $$

Key Concepts

Simplifying Expressions

Simplifying expressions in algebra involves reducing a mathematical expression to its simplest form. This process makes it easier to understand and use in calculations or further problem-solving. In our exercise, the original expression is \[ A(A+7)+4(A^{2}+3A+1) \] To simplify this, we aim to organize and reduce the expression without changing its value. This involves a few key steps such as distribution and combining like terms.
By simplifying, we make expressions less cluttered, improve our understanding, and prepare them for application in equations. The goal is always clarity and reduced complexity.

Distributive Property

The distributive property is one of the fundamental principles in algebra. It states that a term outside of a parenthesis can be distributed to each term inside the parenthesis. In mathematical terms, for any numbers or variables \(a\), \(b\), and \(c\):
\[a(b+c) = ab + ac\]
In our example, this property helps us manage the expression by breaking it down:- For the term \(A(A+7)\), distribute \(A\) to both \(A\) and \(7\), resulting in \(A^2 + 7A\).- For \(4(A^2 + 3A + 1)\), distribute \(4\) to each term inside: \(4A^2 + 12A + 4\).
Understanding the distributive property is crucial for effectively simplifying expressions. It turns seemingly complex expressions into manageable parts.

Like Terms

Terms in an expression are called 'like terms' if they have the exact same variables raised to the same powers. Combining like terms is an essential step in simplifying expressions.
In the expression \(A^2 + 7A + 4A^2 + 12A + 4\):- \(A^2\) and \(4A^2\) are like terms because they both contain the \(A\) variable squared.- \(7A\) and \(12A\) are like terms due to both having the \(A\) variable raised to the first power.
Identifying and combining like terms reduces the complexity of the expression, making it simpler and easier to interpret. It is a straightforward method of aggregation that leads us to a simplified sum.

Combining Terms

Combining terms is the final aggregating step when simplifying an algebraic expression. After identifying like terms, we sum their coefficients to merge them into a single term. This process involves adding or subtracting the coefficients of the terms that are alike.
In the provided expression, we combine:

• \(A^2\) and \(4A^2\) to get \(5A^2\)
• \(7A\) and \(12A\) to compute \(19A\)
• Finally, add the constant \(4\)

Once combined, the expression \[A^2 + 4A^2 + 7A + 12A + 4\] transforms into the simplified form\[5A^2 + 19A + 4\].Combining terms is like tidying up a room, ensuring each piece is in its proper place and clustered efficiently.