Question

Factor. $$7 x^{3}-3 x^{2}$$

Short answer

Answer: The factored form of the polynomial is $$x^2 (7x - 3)$$.

Step-by-step solution

Identify the GCD of coefficients and variable terms

The GCD of the coefficients 7 and -3 is 1. The GCD of the variable terms \(x^3\) and \(x^2\) is \(x^2\). Combining these, the GCD of the given polynomial is \(1x^2\) or simply \(x^2\).

Factor out the GCD from the polynomial

We will now factor out the \(x^2\) from the polynomial $$7 x^{3}-3 x^{2}$$. To do this, divide each term by \(x^2\) as follows: $$x^2 (7x - 3)$$

Simplify the expression

Since the expression is already simplified, the factored form of the polynomial $$7 x^{3}-3 x^{2}$$ is $$x^2 (7x - 3)$$.

Key concepts

Greatest Common Divisor (GCD)

The greatest common divisor, or GCD, is the largest number that can perfectly divide each term in a set of numbers. In polynomial expressions, it can involve coefficients and variable terms. To find the GCD of a polynomial like the one our exercise demonstrates,
follow these steps:

• Identify the coefficients of each term. In our example, the coefficients are 7 and -3.
• Find the GCD of these numbers. Here, 7 and -3 have a GCD of 1, since 1 is the only number that evenly divides both.
• Look at the variables. The variable terms are \(x^3\) and \(x^2\).
• Determine the lowest power of x in these terms, which is \(x^2\).

Combining these, the GCD for the polynomial \(7x^3 - 3x^2\) is \(x^2\). We factor \(x^2\) out to simplify, creating a foundation for further simplification.

Monomials

A monomial is essentially a single-term polynomial. It is composed of a coefficient and variables raised to powers, like \(7x^3\) and \(-3x^2\) in the original exercise.
Monomials have certain key characteristics:

• They consist of only one term without addition or subtraction within that term.
• Each part of the term is either a constant or a product of constants and variables.
• They are straightforward to work with due to their single-term nature.

Understanding monomials is crucial as they form the building blocks of larger polynomial expressions. By mastering how to handle monomials, factoring becomes a much simpler task, as in the case of \(x^2(7x - 3)\), which simplified our original polynomial expression.

Factoring Polynomials

Factoring polynomials involves expressing a polynomial as the product of its simpler building blocks, such as monomials or binomials. The objective when factoring is to make multiplication easier and to simplify complex expressions. Here's how it's done:

• Identify and factor out the GCD. As seen in our task, \(x^2\) was the GCD of \(7x^3 - 3x^2\).
• Once the GCD is factored out, simplify the remainder of the expression. For \(7x^3 - 3x^2\), dividing each term by \(x^2\) gives \(7x - 3\).
• Finally, express the entire polynomial as the product of the GCD and the remaining simplified polynomial. So, the factored form is \(x^2(7x - 3)\).

The factored expression is useful for solving equations, simplifying algebraic expressions, and understanding polynomial behavior. Mastery of factoring transforms complex algebra into more manageable tasks.