Unlocking the Secrets of (6r + 1)(8r + 3): A Deep Dive

The product of (6r + 1)(8r + 3), when expanded and simplified, is 48r² + 26r + 3. This is achieved through the application of the distributive property (often remembered by the acronym FOIL: First, Outer, Inner, Last), a cornerstone of algebraic manipulation.

Deconstructing the Product: From Binomials to Trinomials

Let’s break down how we arrive at this solution. We’re dealing with the multiplication of two binomials: (6r + 1) and (8r + 3). The distributive property dictates that each term in the first binomial must be multiplied by each term in the second binomial.

• First: Multiply the first terms of each binomial: (6r) * (8r) = 48r²
• Outer: Multiply the outer terms of the binomials: (6r) * (3) = 18r
• Inner: Multiply the inner terms of the binomials: (1) * (8r) = 8r
• Last: Multiply the last terms of each binomial: (1) * (3) = 3

Now, we combine these results:

48r² + 18r + 8r + 3

Finally, we simplify by combining like terms (the ‘r’ terms):

48r² + (18r + 8r) + 3 = 48r² + 26r + 3

Thus, the product of (6r + 1)(8r + 3) is indeed 48r² + 26r + 3.

Frequently Asked Questions (FAQs)

Here are some commonly asked questions related to this algebraic expansion, designed to solidify your understanding:

1. What is the distributive property?

The distributive property is a fundamental algebraic rule that allows you to multiply a single term by multiple terms within a set of parentheses. In its simplest form, it’s expressed as a(b + c) = ab + ac. In our example, it’s applied to multiplying two binomials. It’s a cornerstone of simplifying and solving algebraic expressions.

2. Can FOIL be used for trinomials?

No, FOIL is specifically designed for multiplying two binomials (expressions with two terms). For multiplying a binomial by a trinomial or two trinomials, you need to apply the distributive property methodically, ensuring that each term in one expression is multiplied by each term in the other. Using the distributive property ensures that every term is multiplied correctly, irrespective of the number of terms in each set of parenthesis.

3. Is there another method for multiplying binomials?

Besides FOIL, the box method (also known as the grid method) is another excellent technique for multiplying binomials. It involves creating a grid where each row and column is labeled with the terms of the binomials. You then multiply the corresponding row and column labels to fill each cell of the grid. Finally, you add up all the terms within the grid, combining like terms to get the simplified result. The box method is especially helpful for complex expressions with multiple terms.

4. How do you simplify algebraic expressions?

Simplifying algebraic expressions involves combining like terms. Like terms are terms that have the same variable raised to the same power. For instance, 5x and 3x are like terms, but 5x and 3x² are not. To simplify, you add or subtract the coefficients (the numerical part) of the like terms while keeping the variable and its exponent the same.

5. What happens if ‘r’ has a specific value?

If ‘r’ has a specific numerical value, you substitute that value into the simplified expression 48r² + 26r + 3 and perform the arithmetic. For example, if r = 2, then the expression becomes 48(2)² + 26(2) + 3 = 48(4) + 52 + 3 = 192 + 52 + 3 = 247.

6. What if there were a negative sign in front of one of the binomials?

If there were a negative sign in front of one of the binomials, you would need to distribute that negative sign to every term within that binomial before applying the distributive property (FOIL). For instance, if you had -(6r + 1)(8r + 3), you could distribute the negative sign to the first binomial to get (-6r – 1)(8r + 3) and then proceed with the multiplication.

7. Can this method be used for complex numbers?

Yes, the distributive property applies even when dealing with complex numbers. Remember that a complex number is of the form a + bi, where ‘a’ and ‘b’ are real numbers and ‘i’ is the imaginary unit (√-1). The same principles of multiplying and combining like terms apply, remembering that i² = -1.

8. What are the common mistakes to avoid?

Common mistakes include: forgetting to multiply all terms correctly, making errors in sign (especially when dealing with negative numbers), failing to combine like terms properly, and incorrectly applying the order of operations (PEMDAS/BODMAS). Careful attention to detail and practicing regularly can help avoid these pitfalls.

9. What is the difference between an expression and an equation?

An expression is a combination of variables, constants, and operations, without an equals sign. For instance, 48r² + 26r + 3 is an expression. An equation, on the other hand, is a statement that two expressions are equal, containing an equals sign (=). For example, 48r² + 26r + 3 = 0 is an equation.

10. How does this relate to factoring?

Factoring is essentially the reverse process of expanding. While expanding involves multiplying expressions, factoring involves breaking down an expression into its constituent factors. Recognizing patterns in expressions like the one we obtained (48r² + 26r + 3) can help in factoring them back into their original binomial form (6r + 1)(8r + 3).

11. What are some real-world applications of this type of algebra?

This type of algebraic manipulation is used extensively in various fields, including engineering, physics, computer science, and economics. For example, engineers use it to model the behavior of circuits, physicists use it to describe motion and forces, computer scientists use it in algorithm design, and economists use it to analyze market trends.

12. How can I improve my skills in algebraic manipulation?

Practice is key! Work through numerous examples, starting with simpler problems and gradually increasing the complexity. Pay close attention to the rules of algebra, especially the distributive property, order of operations, and combining like terms. Use online resources, textbooks, and seek help from teachers or tutors when needed. Regularly revisiting fundamental concepts will build a solid foundation and improve your skills over time. Remember, consistency is crucial for mastering any mathematical skill.