Question

Use the method of your choice to factor the polynomial completely. Explain your reasoning. $$ 5 x^5-10 x^4-40 x^3 $$

Short answer

The completely factored form of the polynomial \(5x^5 - 10x^4 - 40x^3\) is \(5x^3(x - 4)(x + 2)\.

Step-by-step solution

Identifying the greatest common factor

Observe the given polynomial: \(5x^5 - 10x^4 - 40x^3\). Here, the terms share a common factor of \(5x^3\), which can be factored out to simplify the polynomial.

Factorizing using the greatest common factor

Factor out the greatest common factor \(5x^3\) from all terms, leading to the equation: \(5x^3(x^2 - 2x - 8)\).

Simplifying the polynomial

In the resulting polynomial \(x^2 - 2x - 8\) within the brackets, it can be seen that it is a quadratic equation in standard form that can be simplified into two factors. This equation can be factored into \((x - 4)(x + 2)\). This follows the general format of factorizing a quadratic equation in the form \(ax^2 + bx + c\), where the two factors are \( (x - p)(x - q)\), and \(pq = c\), \(p + q = b\).

Final form of the factored polynomial

The final factored form of the polynomial \((5x^5 - 10x^4 - 40x^3)\) therefore is \(5x^3(x - 4)(x + 2)\)

Key concepts

Greatest Common Factor

To begin factoring polynomials, it's critical to identify the Greatest Common Factor (GCF) among the terms. The GCF is the largest factor that divides each term of the polynomial without a remainder.
For example, in the polynomial \(5x^5 - 10x^4 - 40x^3\), notice each term includes both a numerical coefficient and a power of \(x\).
The numerical GCF of the coefficients \(5\), \(10\), and \(40\) is \(5\). This is the largest number that evenly divides into each coefficient.
The GCF of the variable components is determined by the smallest power of \(x\) that appears in all terms. Here, the smallest power is \(x^3\).
Therefore, the GCF for the entire polynomial is \(5x^3\).

Finding the GCF simplifies the polynomial by dividing each term by this common factor, leading to a simpler expression that retains the original polynomial's structure.

Quadratic Equation

After factoring out the GCF, the remaining polynomial may sometimes form a quadratic equation, which is a specific type of polynomial with a degree of two. In our example, the expression \(x^2 - 2x - 8\) is a standard quadratic form \(ax^2 + bx + c\).
A quadratic equation can often be factored into two binomials, \((x - p)(x + q)\).
To solve this:

• Identify numbers \(p\) and \(q\) such that \(pq = c\) and \(p + q = b\).
• For \(x^2 - 2x - 8\), \(p\) and \(q\) are \(-4\) and \(2\) since \((-4) \, \times \, 2 = -8\) and \((-4) \,+ \, 2 = -2\).

Using this method allows you to break a quadratic polynomial into simpler, multiplied linear expressions, \((x - 4)(x + 2)\).
This makes further simplification and/or solving polynomial equations more manageable.

Polynomial Simplification

Polynomial simplification is the process of breaking down a polynomial into its simplest components or expressions. This often involves factoring, such as extracting the Greatest Common Factor or decomposing quadratics.
For the equation \(5x^5 - 10x^4 - 40x^3\), simplification involves both these forms of factoring.

• First, we factor out the GCF \(5x^3\), resulting in \(5x^3(x^2 - 2x - 8)\).
• Next, we recognize and further simplify the quadratic \(x^2 - 2x - 8\) into factors \((x - 4)(x + 2)\).

Combining both factoring steps transforms the original polynomial into a concise product \(5x^3(x - 4)(x + 2)\).
This form is often easier to interpret, calculate with, or solve as each component is clear and manageable.
Understanding these techniques enhances one's algebraic manipulation skills and facilitates solutions to complex polynomial problems.