Problem 24 - Algebra and Trigonometry

Question

In Exercises 15–58, find each product. $$ (7 x+4)(3 x+1) $$

Step-by-Step Solution

Answer

The product of the binomials \((7 x+4)(3 x+1)\) is \(21x^2 + 19x + 4\).

Step 1: First

Multiply the first terms in each parentheses: \(7x * 3x = 21x^2\).

Step 2: Outside

Multiply the outside terms in each pair of parentheses: \(7x * 1 = 7x\).

Step 3: Inside

Multiply the inside terms in each pair of parentheses: \(4 * 3x = 12x\).

Step 4: Last

Finally, multiply the last terms in each parentheses: \(4 * 1 = 4\).

Step 5: Combine like terms and simplify

Now, combine the outside and inside products since they are like terms which means the same variable raised to the same power. Adding \(7x + 12x\) results in \(19x\), so the final simplified expression is \(21x^2 + 19x +4\).

Key Concepts

FOIL Method

When multiplying two binomials, a straightforward technique called the FOIL Method is often used. FOIL stands for First, Outside, Inside, Last, which represents the order in which you multiply the terms in the binomials. For example, with the expression (7x + 4)(3x + 1),

you follow these steps:

• Multiply the First terms: 7x and 3x to get 21x^2.
• Multiply the Outside terms: 7x and 1 to get 7x.
• Multiply the Inside terms: 4 and 3x to get 12x.
• Multiply the Last terms: 4 and 1 to end up with 4.

By carefully following the FOIL sequence, we can ensure all parts of each binomial are multiplied together, resulting in a polynomial that we can then simplify.

Combining Like Terms

After using the FOIL method, you often end up with an expression that includes 'like terms'. These are terms that have exactly the same variables raised to the same powers. In algebra, we can add or subtract like terms to simplify expressions. For our example, 21x^2 + 7x + 12x + 4,

the like terms are 7x and 12x because they are both multiples of x to the first power. By adding them together, we get 19x, simplifying our polynomial to 21x^2 + 19x + 4. This step is crucial for making expressions more manageable and is often used in algebraic operations.

Algebraic Expressions

An algebraic expression is a mathematical phrase that consists of numbers, variables, and operations. The expression 21x^2 + 19x + 4 is a perfect example of such an expression. It includes variables ( x) raised to different powers, coefficients ( 21, 19, and 4), and the implied addition. These expressions are the bread and butter of algebra and provide a means of representing real-world quantities and their relationships. Understanding how to manipulate these expressions through addition, subtraction, multiplication, and factoring is fundamental to mastering algebra.

Simplifying Polynomials

Simplifying polynomials involves reducing them to their simplest form by performing operations like addition and subtraction on their terms. In the case of our example, we started with the product of two binomials and simplified it to obtain the polynomial 21x^2 + 19x + 4. The simplification process made sure there were no like terms left to combine and that the polynomial was presented in standard form, which typically means having terms in descending order of power. Remember that simplifying not only makes the expression cleaner and easier to understand but is also essential when solving equations or further manipulating algebraic expressions.