Problem 65 - Elementary Algebra

Question

Are the following factored correctly? Check by multiplying. $$ 4 x 2-16 x=4 x(x-4) $$

Step-by-Step Solution

Answer

Yes, it is factored correctly.

Step 1: Expand the Factored Expression

Start with the expression on the right-hand side, which is factored: \(4x(x-4)\). To check if this factoring is correct, first expand it by distributing the \(4x\) across the terms in the parentheses:\[4x(x-4) = 4x \cdot x - 4x \cdot 4\]\[= 4x^2 - 16x\].

Step 2: Compare the Expanded Expression

Now compare the expanded result \(4x^2 - 16x\) with the original left-hand side of the equation \(4x^2 - 16x\). Both expressions are the same, indicating that the factoring of \(4x(x-4)\) was done correctly.

Key Concepts

Multiplying Expressions

When dealing with algebra, multiplying expressions is a foundational skill. At its core, it's about combining terms by multiplying their coefficients and variables. Consider an example where you have to multiply the expression \(4x(x-4)\). Here, you're using the principle that each term inside the parentheses must be multiplied by the expression outside.
Here's how you approach it:

• Multiply the outer term by each inner term: this is called distribution. For example, \(4x\) multiplies both \(x\) and \(-4\).
• For the first multiplication \(4x \cdot x\), you multiply the coefficients (4 and the implied 1 in \(x\)) and the variables \(x \cdot x\) or \(x^1 \cdot x^1 = x^{1+1} = x^2\).
• Next, \(4x \cdot -4\) results in multiplying the coefficients 4 and -4, while \(x\) remains as is.

After carrying out these multiplications, you combine like terms if necessary. This helps in translating factors into expanded expressions.

Distributive Property

The distributive property is a key concept in algebra that helps you multiply a single term by two or more terms within a parenthesis. This principle ensures that every term in the bracket is multiplied by the term outside. Consider any expression of the form \(a(b + c)\). Using the distributive property, it becomes \(ab + ac\). The same principle applies to our original exercise, \(4x(x-4)\).
Here's why the distributive property is crucial:

• It allows for the seamless multiplication of terms across different parts of an expression.
• It's fundamental for both expanding expressions and simplifying complex equations.
• Helps confirm the correctness of factoring, as seen when checking if \(4x(x-4)\) equals the original expression \(4x^2 - 16x\).

By ensuring you correctly apply the distributive property, factoring and simplifying expressions becomes much more manageable.

Variable Expressions

In algebra, variable expressions involve numbers, letters, or both, which represent values in equations. These expressions can vary, taking on different numbers depending on what's being calculated. In our example, \(4x(x-4)\), two variables are at play: \(x\) and each appearance of \(x\) can hold any number in its place.
Understanding variable expressions includes knowing:

• Variables can symbolize unknown values or numbers yet to be determined.
• They allow flexibility in equations, making it possible to solve for various scenarios.
• Combining variables through operations (as seen in \(x\cdot x\)) adheres to specific mathematical rules, like adding exponents with the same base.

When factoring or expanding, recognizing the role of variables helps you transparently see the structure of an expression and to solve it effectively.