Question

Factor by grouping.\(5 x^{3}-10 x^{2}+3 x-6\)

Short answer

The factored form of the expression \(5 x^{3}-10 x^{2}+3 x-6\) by grouping is: \((x-2)(5x^2 + 3)\)

Step-by-step solution

Group the Terms

Rearrange the expression into two groups: \(5 x^{3}-10 x^{2}\) and \(3 x-6\)

Factor Out the Greatest Common Factor From Each Group

From the first group \(5 x^{3}-10 x^{2}\), factor out \(5 x^{2}\) to get \(5 x^{2}(x-2)\). From the second group \(3 x-6\), factor out 3 to get \(3(x-2)\)

Factor by Grouping

Notice that the two groups \(5 x^{2}(x-2)\) and \(3(x-2)\) now have a common factor \(x-2\). Factor this out to get: \((x-2)(5x^2 + 3)\)

Key concepts

Polynomial Factoring

In algebra, polynomial factoring is the process of breaking down a complex polynomial into simpler parts, known as factors, that when multiplied together give back the original polynomial. This technique is akin to finding what ingredients went into a recipe. When dealing with polynomials, such as the one in our exercise, \(5 x^{3}-10 x^{2}+3 x-6\), it resembles a little mathematical puzzle.

Factoring can simplify algebraic expressions and solve equations easier. It's the reverse of expanding and is particularly useful when solving for the zeros of a polynomial. To factor by grouping, which is the method used in our exercise, we look for common factors in grouped terms. However, not all polynomials can be factored easily, and there are different factoring techniques for different types of polynomials. Polynomial factoring is a skill that not only helps in simplifying expressions but also lays the groundwork for more advanced topics like calculus and linear algebra.

Greatest Common Factor

The greatest common factor (GCF) is, simply put, the largest factor that two or more numbers or algebraic terms have in common. It's like finding the biggest piece of the pie that can be evenly served to each person at the table. In the context of algebra, finding the GCF can dramatically simplify expressions by reducing the terms to their simplest form.

Identifying the GCF is a crucial step when factoring by grouping, as done in our example. It involves looking at each group of terms, \(5 x^{3}-10 x^{2}\) and \(3 x-6\), and asking, 'What is the largest expression that can be taken out of each group?' It's an early step in factorization, setting the stage for the rest of the process. Knowing how to find the GCF not only assists with factorization but also encourages a deeper understanding of the structure within algebraic expressions.

Algebraic Expressions

Lastly, let's delve into the world of algebraic expressions, which are like sentences in the language of mathematics where numbers, variables, and operators combine to convey a certain meaning or value. In algebra, these expressions form the backbone of equations and functions. They come in all shapes and sizes, from simple like \(2x+3\) to the more complex like our exercise expression \(5 x^{3}-10 x^{2}+3 x-6\)

An expression becomes especially intersting when attempting to simplify or manipulate it, such as in factoring. Understanding the structure of these expressions—identifying like terms, coefficients, variables, and exponents—is fundamental. When you master the language of algebraic expressions, you unlock the ability to manipulate and solve a wide array of mathematical problems.