Problem 62 - Algebra A Combined Function

Question

Mixed Practice Multiply. $$ (6 a+7)(6 a+5) $$

Step-by-Step Solution

Answer

The product is \(36a^2 + 72a + 35\).

Step 1: Understand the Problem

We are tasked with multiplying two binomials: \((6a + 7)(6a + 5)\). This requires using the distributive property, also known as the FOIL method (First, Outer, Inner, Last).

Step 2: Apply the FOIL Method

Use FOIL to multiply each term in the first binomial by each term in the second binomial:- First: Multiply the first terms: \(6a \times 6a = 36a^2\)- Outer: Multiply the outer terms: \(6a \times 5 = 30a\)- Inner: Multiply the inner terms: \(7 \times 6a = 42a\)- Last: Multiply the last terms: \(7 \times 5 = 35\)

Step 3: Combine Like Terms

Add the resulting terms from the FOIL method together: \(36a^2 + 30a + 42a + 35\).Combine the like terms, which are the \(a\) terms:\(30a + 42a = 72a\).

Step 4: Write the Final Expression

Write down the polynomial expression combining all the simplified terms:\(36a^2 + 72a + 35\).

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that deals with how multiplication interacts with addition. When applied to binomial multiplication, it involves distributing each term in one binomial across each term in the other binomial. For example, if you have

• \((a + b)(c + d)\), each term from the first binomial must be multiplied with each term of the second binomial.

This operation ensures that all possible products are accounted for. It's like ensuring no tile is left unpainted when spreading paint over a grid.
Here’s how it visually works:

• First term from the first binomial multiplies each term in the second binomial.
• Second term from the first binomial does the same.

This comprehensive multiplication is the essence of the distributive property, a critical tool for handling polynomial expressions.

FOIL Method

The FOIL method is a specific application of the distributive property that simplifies multiplying two binomials. This acronym stands for First, Outer, Inner, Last and helps remember the order of operations:

• First: Multiply the first terms of each binomial together.
• Outer: Multiply the outermost terms in the product.
• Inner: Multiply the inner terms.
• Last: Finally, multiply the last terms of each binomial.

Here's a breakdown of how it applies to our example \((6a + 7)(6a + 5)\):

• First: \(6a imes 6a = 36a^2\)
• Outer: \(6a imes 5 = 30a\)
• Inner: \(7 imes 6a = 42a\)
• Last: \(7 imes 5 = 35\)

This method ensures you don't miss any term when multiplying the binomials, providing a structured way to expand them fully.

Like Terms

In algebra, like terms are terms that have the same variables raised to the same power. Only coefficients of like terms can differ. For example, in a polynomial expression,
terms like \(30a\) and \(42a\) are considered like terms because they both include the variable \(a\) to the first power.
Combining like terms involves adding or subtracting their coefficients while retaining the identical variable part. For our binomial multiplication result:

• The like terms are \(30a\) and \(42a\). You combine them like this:
• \(30a + 42a = 72a\)

This simplification process is crucial for reducing a polynomial to its simplest form, making it easier to interpret and work with.

Polynomial Expression

A polynomial expression is a mathematical phrase composed of variables, coefficients, and exponents. These expressions involve addition, subtraction, and multiplication but never division by a variable. Polynomials can have one or many terms.
The degree of a polynomial is determined by the highest power of the variable present.
Consider the final expression we derived: \(36a^2 + 72a + 35\). This is a

• degree 2 polynomial (quadratic) because the highest power of \(a\) is 2.

Terms within a polynomial are distinct based on the power of the variable they involve. Understanding the structure of polynomial expressions is essential for various branches of mathematics, from simple algebra to complex calculus.