Problem 80 - Algebra 1

Question

Find the product. $$(x+4)(3 x-7)$$

Step-by-Step Solution

Answer

So the product \((x+4)(3x-7)\) equals to \(3x^2 + 5x - 28\)

Step 1: Distribution of terms

Write the expression as \((x+4)(3x-7)\). Now, distribute each term of the first polynomial with each term of the second polynomial. Thus, we have four separate multiplication operations: \(x * 3x\), \(x * -7\), \(4 * 3x\), \(4 * -7\)

Step 2: Evaluate multiplication operations

Multiply each pair of terms to get: \(3x^2\), \(-7x\), \(12x\), \(-28\)

Step 3: Combine like terms

Combine the middle two terms which are like terms to simplify the expression: \(3x^2 + 5x - 28\)

Key Concepts

Distributive Property

The distributive property is a fundamental rule in algebra that helps us simplify complex expressions. Imagine you have two polynomials, like \(x+4\) and \(3x-7\), and you want to find their product. The distributive property tells us how to handle the multiplication of every term in the first polynomial with every term in the second one.

Here's how it breaks down:

• Multiply the first term of the first polynomial (\(x\)) by each term in the second polynomial. This gives us \(x * 3x\) and \(x * -7\).
• Repeat this step for the second term of the first polynomial (\(4\)), multiplying it by each term in the second polynomial. This results in \(4 * 3x\) and \(4 * -7\).

By carrying out these operations, we've distributed every term across the polynomials, setting the stage for forming a new polynomial by combining results. This step ensures that multiplication respects both polynomials' structures.

Combining Like Terms

After using the distributive property to expand polynomials, the next crucial step is combining like terms. Like terms are those that have the same variables raised to the same power. They can be added or subtracted together because they essentially represent the same type of object.

Let's look at the expression derived from the distributive property: \3x^2 - 7x + 12x - 28\. Notice the middle terms, \-7x\ and \12x\? They are like terms because both have the variable \(x\) raised to the first power.

• Simply add or subtract their coefficients: \(-7+12=5\), giving us \5x\.

Combining these terms produces the simplified polynomial: \3x^2 + 5x - 28\. By merging like terms, we simplify and reduce expressions, making them easier to understand and work with.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operations. They form the foundation of algebra and are crucial for solving equations. In our exercise, \(x+4\) and \(3x-7\) are algebraic expressions that consist of:

• Variables: like \(x\), which stand for unknown values.
• Constants: like \(4\) and \(-7\), representing fixed values.
• Operations: addition, multiplication, and subtraction.

These expressions are combined using the distributive property to create a new algebraic expression. Understanding how to manipulate these expressions allows you to solve complex algebraic problems. This skill is foundational for higher mathematics and various applications in fields such as engineering and physics.