Question

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

Short answer

The factors of the given polynomial \(x^{3}-x^{2}+2 x-2\) by grouping are \((x - 1)\) and \((x^2 + 2)\). So, \(x^{3}-x^{2}+2 x-2 = (x - 1)(x^2 + 2)\).

Step-by-step solution

Divide into Groups and Factor

Break the expression into two groups and factor out the greatest common factor from each. Our polynomial is \(x^{3}-x^{2}+2 x-2\), it can be grouped as: \((x^{3}-x^{2})+(2 x-2)\), and factoring we get \(x^2(x - 1) + 2(x - 1)\).

Factor by Grouping

Now that we've factored out the common factors from each group, we can see that the expression within the brackets (x - 1) is common. We can further factorize by taking it as a common factor: \((x - 1)(x^2 + 2)\).

Verify the Results

Expand the factors to verify the results. Multiply \((x - 1)\) with \((x^2 + 2)\) and check if the result matches with the original polynomial. It should if the steps were followed correctly.

Key concepts

Factoring Polynomials

When you're working with polynomials, factoring is a key skill that helps simplify complex algebraic expressions. Factoring involves breaking down a polynomial into simpler components—known as factors—that, when multiplied together, return the original polynomial. It's akin to splitting a number into a multiplication of its prime factors.

For instance, consider the quadratic polynomial \(x^2 - 5x + 6\). Factored form of this polynomial would be \( (x - 2)(x - 3) \), since \( (x - 2) \times (x - 3) = x^2 - 5x + 6\). In algebra, factoring polynomials is essential not only for simplification but also for solving equations and understanding graphs of polynomial functions.

Greatest Common Factor

The greatest common factor (GCF), also known as the greatest common divisor, is the largest factor shared by two or more numbers or terms. In algebra, locating the GCF of different terms within a polynomial is pivotal in the factoring process. For example, the GCF of the numbers 12 and 18 is 6, as it's the largest number that divides both evenly.

In algebraic terms, if you have a polynomial like \( 6x^2 + 18x \), the GCF is \(6x\). Finding the GCF is a crucial first step in factoring polynomials because once identified, it can be factored out, simplifying the rest of the expression. This often leads to a more manageable problem and paves the way to further factorization.

Polynomial Factorization

Polynomial factorization is the process of decomposing a polynomial into a product of simpler polynomials or factors that, when multiplied, give back the original polynomial. There are several methods for factoring polynomials, and one such technique is 'factor by grouping' which you encounter when dealing with quadratics or more complex polynomials.

To factor by grouping, we first split the polynomial into groups of terms that share a common factor. Then, we factor out the GCF from each group separately. Finally, if the same factor appears in each group, we can factor that out as well, simplifying our expression to a product of factors.

Algebraic Expressions

Algebraic expressions are mathematical phrases combining numbers, variables, and operational symbols that do not include an equality sign. They can range from simple, like \(3x + 2\), to extremely complex with multiple terms and powers, such as \(6x^4 - 3x^3 + 2x^2 - x + 1\).

Algebraic expressions are the building blocks of algebra and are used to represent real-world scenarios mathematically. When you factor algebraic expressions, you simplify complex problems and glean more information about the relationships between the variables within the expression. This simplification can lead to solving for unknown variables and better understanding the expression's graph.