Problem 75 - Algebra A Combined Function

Question

Multiply. $$ (a+3 b)(9 a-4 b) $$

Step-by-Step Solution

Answer

The product is \(9a^2 + 23ab - 12b^2\).

Step 1: Distribute the First Term

Start by distributing the first term from the first binomial, \(a\), to each term in the second binomial \((9a - 4b)\). This gives us: \[ a(9a - 4b) = a \cdot 9a - a \cdot 4b = 9a^2 - 4ab \]

Step 2: Distribute the Second Term

Now distribute the second term from the first binomial, \(3b\), to each term in the second binomial \((9a - 4b)\). This gives us: \[ 3b(9a - 4b) = 3b \cdot 9a - 3b \cdot 4b = 27ab - 12b^2 \]

Step 3: Combine Like Terms

Now, combine all the terms from the previous steps. The expression becomes:\[ 9a^2 - 4ab + 27ab - 12b^2 \]Combine the like terms \(-4ab\) and \(+27ab\): \[ 9a^2 + 23ab - 12b^2 \]

Step 4: Write the Final Expression

The final expression is \(9a^2 + 23ab - 12b^2\). Make sure all terms are properly simplified and no further simplifications are possible.

Key Concepts

Binomial Distribution

Let's begin by exploring the concept of binomial distribution, which is a crucial aspect of multiplying polynomials. Binomial distribution refers to the process of multiplying two binomials together. A binomial is simply a polynomial with two terms, like \(a + 3b\) or \(9a - 4b\). When we multiply these two, we start by taking each term from the first binomial and multiplying it by every term in the second binomial one by one. This is also known as the distributive property.

• First, distribute the first term from the first binomial: \(a \times (9a - 4b)\).
• Then, distribute the second term from the first binomial: \(3b \times (9a - 4b)\).

By carefully following these steps, you work through all possible combinations of terms in the binomials, ensuring nothing is missed. Think of it as a way to navigate through and organize each possible multiplication, making sure to apply each term in one binomial to every term in the other.

Like Terms

Understanding and combining like terms is essential for simplifying expressions in polynomial multiplication. Like terms are terms in an expression that have identical variable parts raised to the same power.
For example, in the expression \(9a^2 - 4ab + 27ab - 12b^2\), the terms \(-4ab\) and \(27ab\) are like terms because they both contain the product \(ab\). When combining like terms, only their coefficients are added or subtracted.

• Like terms: \(-4ab\) and \(27ab\).
• Combine to form: \(23ab\) (since \(-4 + 27 = 23\)).

By identifying and combining like terms, you ensure a simpler and cleaner expression. This also prepares the expression for further operations or simplifies it as a final answer. It makes the polynomial more readable and easy to interpret.

Simplification

The final phase of polynomial multiplication is simplification. Simplification involves combining like terms and ensuring that the expression is as reduced as possible. This step verifies that no further operations are needed and you have reached the simplest form of the polynomial.
In our expression \(9a^2 + 23ab - 12b^2\), simplification confirms that:

• The terms \(9a^2\), \(23ab\), and \(-12b^2\) are all distinct and cannot be combined further.
• There are no common factors to factor out.
• All like terms, if any, have been combined earlier.

Simplification ensures mathematical precision and clarity, eliminating unnecessary complexities in expressions. By maintaining a clear and concise expression, it makes understanding the results and their implications significantly easier for students and educators alike.