Problem 84 - Elementary Algebra

Question

For the following problems, perform the multiplications and combine any like terms. $$ (6+a)(4+a) $$

Step-by-Step Solution

Answer

Question: Multiply the two binomials (6+a)(4+a) and simplify the result. Answer: (6+a)(4+a) = 24+10a+a^2

Step 1: Expand using distributive property (FOIL)

To multiply the two binomials \((6+a)(4+a)\), we will apply the FOIL method which stands for First, Outer, Inner, and Last. Multiply the terms in this order: First: \(6\times4\) Outer: \(6\times a\) Inner: \(a\times4\) Last: \(a\times a\) Now, we can write down the product of these terms: $$(6+a)(4+a) = 6\times4 + 6\times a + a\times4 + a\times a$$

Step 2: Simplify the result

Now we simplify the products and combine any like terms: $$(6+a)(4+a)= 24 + 6a + 4a + a^2$$ We can combine the like terms \(6a\) and \(4a\): $$= 24 + (6a + 4a) + a^2$$ $$= 24 + 10a + a^2$$ The final simplified expression is: $$(6+a)(4+a)=24+10a+a^2$$

Key Concepts

Distributive Property

The distributive property is a simple yet powerful tool used in mathematics, especially while dealing with polynomial multiplication. It helps in breaking down complex algebraic expressions into simpler ones. When you multiply an expression like

• \((6+a)(4+a)\), you are actually distributing each term in one binomial to every term in the other binomial.

This is expressed algebraically as:
\[a(b+c) = ab + ac\].
This means that every term inside the parentheses is multiplied by the term outside the parentheses. In the case of multiplying two binomials, like

• \((6+a)(4+a)\),

you apply the distributive property twice, ensuring all terms get multiplied with one another.

FOIL Method

The FOIL method is a specific application of the distributive property that makes multiplying two binomials straightforward. FOIL stands for First, Outer, Inner, and Last, which refers to the positions of the terms in the binomials that you multiply.

• First: Multiply the first terms of each binomial, here it is \(6 \times 4 = 24\).
• Outer: Multiply the outer terms, which is \(6 \times a = 6a\).
• Inner: Multiply the inner terms, giving \(a \times 4 = 4a\).
• Last: Multiply the last terms, resulting in \(a \times a = a^2\).

After calculating each part, these resultant products

• \(24, 6a, 4a, a^2\)

are added together, combining like terms, to achieve the final result.

Combining Like Terms

After using the FOIL method, the expression should be simplified by combining like terms. Like terms are terms that have identical variable parts.

• In \(24 + 6a + 4a + a^2\), the like terms are \(6a\) and \(4a\) because both contain the variable \(a\).

By adding these coefficients, you simplify the expression:
\(6a + 4a\) results in \(10a\).
So, the expression simplifies to

• \(24 + 10a + a^2\).

Always remember to sort it in standard algebraic form \(ax^2 + bx + c\), even if the terms are already in correct order.

Binomials

A binomial is an algebraic expression that consists of exactly two terms. These two terms can include numbers, variables, or their products.
An example of a simple binomial is

• \((6 + a)\) or \((4 + a)\).

When multiplying binomials, it involves applying methods such as the distributive property or the FOIL method to simplify their products.
Binomials are fundamental building blocks in algebra, often forming the basis for more complex polynomials.
In algebra, understanding how to work with binomials is crucial because it leads to the ability to multiply, factor, and simplify more complicated expressions. It's essential to gain comfort with binomials to advance in algebraic concepts.