Problem 58 - Algebra A Combined Function

Question

Mixed Practice Multiply. $$ (a+5)(a+7) $$

Step-by-Step Solution

Answer

The expanded form is \(a^2 + 12a + 35\).

Step 1: Apply the Distributive Property

We begin by applying the distributive property, also known as the FOIL method for binomials. This involves multiplying each term in the first binomial with each term in the second. Thus:\((a+5)(a+7) = a(a+7) + 5(a+7)\).

Step 2: Distribute Each Term

Next, distribute across each set of parentheses independently:\[a(a+7) = a \cdot a + a \cdot 7 = a^2 + 7a\]\[5(a+7) = 5 \cdot a + 5 \cdot 7 = 5a + 35\]

Step 3: Combine Like Terms

Now, combine all terms obtained from the distribution:\[a^2 + 7a + 5a + 35\]Combine the like terms (\(7a\) and \(5a\)): \[a^2 + 12a + 35\].

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that simplifies expressions by distributing multiplication over addition or subtraction. In essence, it tells us that multiplying a single term by a binomial involves multiplying each component of the binomial separately by the single term.
For instance, consider the expression \(a(a+7)\). Using the distributive property, we break it down as follows: \((a \cdot a) + (a \cdot 7)\), giving us the expression \(a^2 + 7a\).

This property is not limited to numbers but also applies to variables and terms in an algebraic context. Understanding and applying the distributive property is crucial because it lays the groundwork for other operations, such as multiplying polynomials, and is a vital step when using the FOIL method.

FOIL Method

The FOIL method is a specific application of the distributive property used exclusively for multiplying two binomials. It stands for First, Outer, Inner, and Last, referring to the order of multiplying the terms in the two binomials. This method helps ensure all terms are accounted for during multiplication.
Let's break it down with the example \(a+5\) and \(a+7\):

• First: Multiply the first terms in each binomial: \(a \cdot a = a^2\).
• Outer: Multiply the outer terms: \(a \cdot 7 = 7a\).
• Inner: Multiply the inner terms: \(5 \cdot a = 5a\).
• Last: Multiply the last terms: \(5 \cdot 7 = 35\).

Combining all these, we reach \(a^2 + 7a + 5a + 35\). By mastering this method, you can efficiently and accurately multiply binomials, a vital skill in algebra and beyond.

Combining Like Terms

In algebra, simplifying expressions is a common task, and combining like terms is a key component of this process. Like terms are terms that share the same variable part and exponent. For instance, in the expression \(7a + 5a\), both terms are like terms because they contain the variable \(a\) raised to the power of one.
To combine like terms, you merely add or subtract their coefficients. In our example, combine \(7a\) and \(5a\) to obtain \(12a\).
Let's consider the full expression from our solution \((a^2 + 7a + 5a + 35)\). After identifying like terms—

• Combine \(7a\) and \(5a\) to result in \(12a\).

This simplification results in the expression \(a^2 + 12a + 35\). Mastering this technique streamlines solving and understanding algebraic expressions, providing clarity and leading to more intuitive results.