Problem 97 - Elementary Algebra

Question

Problem: Simplify: \((4 x-1)(6 x+1)\) $$ \text { Incorrect Answer: } \begin{aligned} &(4 x-1)(6 x+1) \\ &=4 x(6 x)+4 x(1)-1(6 x)-1(1) \\ &=24 x^{2}+4 x-6 x-1 \\ &=24 x^{2}+2 x-1 \end{aligned} $$

Step-by-Step Solution

Answer

24x^2 - 2x - 1

Step 1: Apply the distributive property (FOIL method)

Use the distributive property to expand the expression \( (4x-1)(6x+1) \). FOIL stands for First, Outer, Inner, Last. Multiply the terms in this order.

Step 2: Multiply the first terms

Multiply the first terms in each binomial: \( 4x \times 6x = 24x^2 \).

Step 3: Multiply the outer terms

Multiply the outer terms: \( 4x \times 1 = 4x \).

Step 4: Multiply the inner terms

Multiply the inner terms: \( -1 \times 6x = -6x \).

Step 5: Multiply the last terms

Multiply the last terms: \( -1 \times 1 = -1 \).

Step 6: Combine like terms

Add all the products together: \( 24x^2 + 4x - 6x - 1 \). Combine the like terms \(4x - 6x = -2x \) to get the final simplified form \( 24x^2 - 2x - 1 \).

Key Concepts

FOIL method

The FOIL method is a cool trick to remember when multiplying two binomials. FOIL stands for First, Outer, Inner, and Last, and it helps ensure you multiply every term in the first binomial by every term in the second. This method follows a specific order to make sure no term is missed.
First, multiply the first terms of each binomial. In our problem, that's multiplying 4x from \(4x-1\) and 6x from \(6x+1\), giving us \(24x^2\).
Next, move to the outer terms. Multiply 4x (the term on the outermost part of the first binomial) by 1 (the term on the outermost part of the second binomial). You'll get \(4x\).
Then, multiply the inner terms which are -1 and 6x. This gives us \(-6x\).
Finally, multiply the last terms in each binomial: -1 and 1, resulting in \(-1\).
Put all these pieces together: \(24x^2 + 4x - 6x - 1\). Understanding FOIL helps prevent mistakes and ensures that every part of the binomials is considered.

Distributive Property

The distributive property is the foundation of the FOIL method. It states that \a(b + c) = ab + ac\. In simple terms, you multiply everything inside the parentheses by what's outside.
Using our exercise as an example, we distribute each term in \(4x-1\) to each term in \(6x+1\).
We perform the following multiplications:

• First terms: 4x \times 6x = 24x^2
• Outer terms: 4x \times 1 = 4x
• Inner terms: -1 \times 6x = -6x
• Last terms: -1 \times 1 = -1

Break complex expressions into smaller parts and then combine them back together. This method is very powerful in algebra, making it easier to simplify and solve polynomial expressions.

Combine Like Terms

Combining like terms is crucial in simplifying polynomials. It's easy to overlook, but it helps in reducing your answer to its simplest form. A 'like term' is any term that shares the same variable raised to the same power.
After expanding the binomials using FOIL and distributive property, you often end up with multiple terms that can be combined. In our example, after applying FOIL, we get \(24x^2 + 4x - 6x - 1\).
Notice how we have \(4x\) and \(-6x\)? Since they are like terms, we combine them:
\(4x - 6x = -2x\).
So, our expression simplifies to \(24x^2 - 2x - 1\). Combining like terms makes your answer cleaner and ensures that it's fully simplified.

Polynomial Simplification

Polynomial simplification is about making expressions easier to read and work with. You achieve this by using various algebraic techniques like the FOIL method, the distributive property, and combining like terms.
Let's take a fresh look at our exercise. First, we applied FOIL to distribute the terms and obtained \(24x^2 + 4x - 6x - 1\).
Following that, we identified and combined like terms (4x and -6x), giving us \(-2x\).
Finally, we simplified the expression to \(24x^2 - 2x - 1\).
The process of polynomial simplification is essential in algebra because it transforms complex expressions into simpler forms, making solving equations and performing further algebraic operations more manageable.
Remember, practice makes perfect. Reviewing each technique can build your confidence and help you master polynomial simplification.