Question

Factor completely: \(2 x^{4}+6 x^{3}-50 x^{2}-150 x\)

Short answer

The polynomial factors to [ 2x(x + 3)(x - 5)(x + 5) ]

Step-by-step solution

- Factor out the Greatest Common Factor (GCF)

Identify and factor out the greatest common factor (GCF) in the polynomial. The GCF of the terms in the polynomial is 2x. Thus, factor 2x out of each term in the polynomial: [ 2x(x^3 + 3x^2 - 25x - 75) ]

- Factor by grouping

Group the terms in pairs to facilitate factoring by grouping. Group the first two terms and the last two terms: [ 2x[(x^3 + 3x^2) + (-25x - 75)] ]

- Factor out the common factor in each group

Factor out the common factor from each group: [ 2x[x^2(x + 3) - 25(x + 3)] ]

- Factor out the common binomial factor

Now, factor out the common binomial factor [ (x + 3) ]. This gives: [ 2x(x + 3)(x^2 - 25) ]

- Factor the difference of squares

Recognize that [ x^2 - 25 ] is a difference of squares and can be factored further: [ x^2 - 25 = (x - 5)(x + 5) ]. Thus, the polynomial can be written as: [ 2x(x + 3)(x - 5)(x + 5) ]

Key concepts

Greatest Common Factor

The greatest common factor (GCF) is the highest number or algebraic expression that divides each term in a polynomial without leaving a remainder.
Identifying and factoring out the GCF is often the first step in simplifying a polynomial.
For the polynomial in our example, the GCF is 2x.
To find the GCF, look for the largest number and the highest power of any variable that appears in each term.
Here’s how you do it:

• List the factors of each term: for instance, for 2x⁴, the factors are 1, 2, x, x², x³, x⁴.
• Identify the common factors: The common factors for all terms include 2 and x.

Factoring out the GCF simplifies the polynomial and sets up the next steps. In this case, factoring out 2x from 2x⁴ + 6x³ − 50x² − 150x yields 2x(x³ + 3x² − 25x − 75).

Factoring by Grouping

Factoring by grouping is particularly useful for polynomials with four terms.
It involves grouping terms that have common factors.
In our example, the polynomial 2x(x³ + 3x² − 25x − 75) can be grouped to make factoring simpler.
Here’s how you proceed:

• Group terms into pairs: (x³ + 3x²) and (−25x − 75).
• Factor out the common factor from each pair: From (x³ + 3x²) you factor x², and from (−25x − 75), you factor -25.

After factoring out the common terms, you get: 2x[x²(x + 3) − 25(x + 3)]. Notice how (x + 3) is common in both groups.

Difference of Squares

A difference of squares refers to a binomial formed as a² - b².
This can be factored into (a + b)(a - b).
In our example, we have x² - 25, which is a difference of squares:

• Identify a and b: Here, a is x and b is 5 because 25 is 5².
• Factor the binomial: x² - 25 becomes (x - 5)(x + 5).

Factoring the difference of squares simplifies the polynomial even further. So, 2x(x + 3)(x² - 25) becomes 2x(x + 3)(x - 5)(x + 5).

Algebraic Expressions

Algebraic expressions include variables, constants, and arithmetic operations like addition, subtraction, multiplication, and division.
Polynomials are a specific type of algebraic expression.
Understanding the structure and components of algebraic expressions is essential.
In our example, 2x⁴ + 6x³ - 50x² - 150x is an algebraic expression where:

• 2x⁴, 6x³, -50x², and -150x are terms.
• 2, 6, -50, and -150 are coefficients.
• x is the variable.

Breaking down each part makes it easier to apply factoring techniques and solve the expression step by step.