Question

Factor the polynomial completely. \(5 x^3-15 x^2\)

Short answer

The factored form of the polynomial \(5x^3 - 15x^2\) is \(5x^2(x - 3)\).

Step-by-step solution

Identify Common Factors

Looking at the terms of the polynomial \(5x^3 - 15x^2\), we can see that each term has a factor of 5 and \(x^2\). Both terms are divisible by these factors.

Factor out the Common Factor from each term

We divide each term by \(5x^2\). This gives us \(5x^3/5x^2 - 15x^2/5x^2\). After dividing, we get \(x - 3\).

Write the final factored form

We then rewrite the original polynomial as the product of the common factor and the result from Step 2. Thus the factored form is \(5x^2(x - 3)\).

Key concepts

Common Factors

When dealing with polynomials, identifying common factors is an essential first step in the factorization process. A common factor is a number or algebraic expression that can evenly divide each term in the polynomial.
For example, in the polynomial \(5x^3 - 15x^2\), the terms have common factors of 5 and \(x^2\).

• Number: The greatest common divisor (GCD) of the coefficients, in this case, 5, can be factored out from both coefficients of the terms.
• Variable: Look for the smallest power of the variable present in all the terms, here it is \(x^2\).

Factoring out these common factors simplifies the polynomial by reducing the powers of variables and coefficients, making it easier to work with. By performing this step, a more manageable expression is obtained.

Factoring Techniques

There are various factoring techniques used to simplify algebraic expressions, and choosing the right one depends on the structure of the polynomial.
In our example of \(5x^3 - 15x^2\), we employed the method of factoring by common factors:

• Factor by grouping: Useful when the polynomial can be divided into groups, each having a common factor.
• Factoring trinomials: Often used for polynomials of three terms (trinomials), particularly when they resemble a perfect square or have a specific form that matches known factorization patterns.
• Prime Factorization: This involves breaking down coefficients into prime numbers, which can sometimes reveal additional factors.

By dividing each term of \(5x^3 - 15x^2\) by the identified common factor \(5x^2\), we simplify the polynomial into its factored form: \(5x^2(x - 3)\). This systematic approach to laying out the problem helps to simplify and understand the polynomial's structure.

Algebraic Expressions

An algebraic expression is a mathematical phrase involving numbers, variables, and operators. They form the basis of how we represent and solve problems using algebra. Polynomials, like \(5x^3 - 15x^2\), are a type of algebraic expression comprising of terms with variables raised to whole number powers.
The term "expression" implies a combination of:

• Coefficients: Numerical factors multiplying variables, such as 5 and -15 in the given polynomial.
• Variables: Symbols (like \(x\)) that represent numbers, making the expression general and applicable in various situations.
• Exponents: Indicate power to which a variable is raised.

Simplifying algebraic expressions through techniques like factoring makes solving equations and analyzing relationships more effective. Understanding how to manipulate these expressions is a crucial skill in algebra, which ultimately leads to solving for unknowns or simplifying complex equations.