Problem 55 - Algebra 2

Question

Simplify. $$ (2 x+4)(7 x-1) $$

Step-by-Step Solution

Answer

The simplified expression is \(14x^2 + 26x - 4\).

Step 1: Apply the Distributive Property

To simplify the expression \((2x + 4)(7x - 1)\), start by applying the distributive property (also called FOIL method for binomials).Multiply the first terms: \(2x \times 7x = 14x^2\).Multiply the outer terms: \(2x \times -1 = -2x\).Multiply the inner terms: \(4 \times 7x = 28x\).Multiply the last terms: \(4 \times -1 = -4\).

Step 2: Combine Like Terms

Now, combine the like terms obtained from the distribution.The expression becomes: \[14x^2 + (-2x) + 28x - 4.\]Simplify it by combining the \(-2x\) and \(28x\):\[14x^2 + 26x - 4.\]

Key Concepts

Algebraic Expressions

Algebraic expressions are mathematical phrases that consist of numbers, variables, and operators, such as addition, subtraction, multiplication, and division.
They are essential in algebra because they allow us to describe various mathematical situations and relationships.
An example of an algebraic expression is the one given in the original exercise: \((2x + 4)(7x - 1)\). This expression contains:

• Variables: Represented by letters like \(x\), these variables hold the place for numbers that can change.
• Coefficients: Numbers that multiply the variables, such as 2 in \(2x\).
• Constants: Fixed values, like 4 and -1 in the expression.

Understanding how to manipulate algebraic expressions is key to solving many algebra problems, especially when learning to simplify them.

Combining Like Terms

Combining like terms is a technique used to simplify algebraic expressions by merging terms that have the same variables raised to the same power.
This process makes an expression easier to understand and solve. Like terms in an algebraic expression have identical variable parts, such as \(-2x\) and \(28x\) in the original exercise. When combining these terms, you simply add or subtract their coefficients.

• In this case, we add the coefficients: \(-2 + 28 = 26\).

Therefore, the expression \(14x^2 + (-2x) + 28x - 4\) simplifies to \(14x^2 + 26x - 4\).
Combining like terms is especially useful after expanding expressions using methods like the distributive property. It helps reduce the expression to its simplest form.

Binomials

Binomials are algebraic expressions that contain exactly two terms.
These expressions play a crucial role in many areas of algebra, including polynomials and quadratic equations. In the exercise provided, both \((2x + 4)\) and \((7x - 1)\) are examples of binomials. Characteristics of binomials include:

• Having two distinct terms, each potentially containing a variable, such as \(2x\) and \(7x\).
• The capability to combine and multiply with other binomials or polynomials using various methods.

When you encounter a problem involving binomials, such as multiplying two binomials, you often employ methods like the FOIL method to simplify or expand these expressions.
Mastering binomials can make solving complex equations more manageable.

FOIL Method

The FOIL method is a specific technique used for multiplying two binomials. The name FOIL stands for First, Outer, Inner, Last, referring to the terms you need to multiply:

• First: Multiply the first term in each binomial.
• Outer: Multiply the outer terms of the binomials.
• Inner: Multiply the inner terms.
• Last: Multiply the last term in each binomial.

In the expression given, \((2x + 4)(7x - 1)\), applying the FOIL method involves:

• First: \(2x \times 7x = 14x^2\)
• Outer: \(2x \times -1 = -2x\)
• Inner: \(4 \times 7x = 28x\)
• Last: \(4 \times -1 = -4\)

These results are then combined to form the expanded expression \(14x^2 + (-2x) + 28x - 4\).
Learning the FOIL method provides a systematic way to multiply binomials, a vital skill in algebra. It ensures you consider all necessary terms when simplifying expressions, leading to correct solutions.