Question

Factor each polynomial completely. If the polynomial cannot be factored, say it is prime. $$ x^{3}+8 x^{2}-20 x $$

Short answer

x(x + 10)(x - 2)

Step-by-step solution

Identify Common Factors

First, look for any common factors among the terms in the polynomial. Each term in the polynomial \bx^{3}+8x^{2}-20xcan be divided by x. Extract x from each term:\[ x(x^{2}+8x-20)\]

Factor the Quadratic Expression

Now, focus on factoring the quadratic expression inside the parentheses: \[x^{2}+8x-20\]Look for two numbers that multiply to -20 and add up to 8. Those numbers are 10 and -2:\[ x(x + 10)(x - 2) \]

Key concepts

Factoring Polynomials

To factor a polynomial means to express it as the product of simpler polynomials. This process is often used to simplify equations, solve for variables, or understand the roots of the polynomial. For example, consider the polynomial given: \( x^{3}+8 x^{2}-20 x \). The goal is to break it up into simpler factors that can be multiplied together to give the original polynomial.

Factoring polynomials can involve several techniques depending on the type of polynomial you are dealing with:

• Factoring out common factors
• Factoring by grouping
• Factoring quadratic expressions
For our polynomial \( x^{3}+8 x^{2}-20 x \), we will use a combination of these techniques.

Common Factors

The first step in factoring any polynomial is to check for common factors. A common factor is a number or variable that evenly divides each term of the polynomial. In the given polynomial, all terms include the variable \( x \). Thus, we can factor out \( x \) from every term:
Extracting \( x \) from \( x^{3}+8 x^{2}-20 x\), we get:
\[ x(x^{2}+8x-20) \]
This simplifies the polynomial and makes it easier to work with the remaining expression inside the parentheses.

Quadratic Expressions

After extracting the common factor, we are left with a quadratic expression inside the parentheses:\( x^{2}+8 x-20 \). A quadratic expression is a polynomial of degree 2, usually written in the form \( ax^{2}+bx+c\). Factoring quadratic expressions usually involves finding two numbers that multiply to give the product of \( a \times c \) and add to give \( b \).

In this case, we need to find two numbers that multiply to \(-20\) (as \( a = 1 \) and \( c = -20 \)) and add up to \( 8 \) (the value of \( b \)).

Those two numbers are 10 and -2 because:

• 10 \times -2 = -20
• 10 + (-2) = 8

Factoring by Grouping

Factoring by grouping is another useful method when dealing with polynomials that have four or more terms. It is not always required, but understanding this technique can make factoring easier in some cases. Here, factoring by grouping isn't necessary because our quadratic expression factors neatly.

For our quadratic expression \( x^{2} + 8x - 20 \), we can directly write it as:
\[ x^{2} + 8x - 20 = (x + 10)(x - 2) \]
Finally, combining this with the common factor we pulled out earlier, we get the fully factored polynomial:
\[ x(x + 10)(x - 2) \]
This is the simplest factorization of the given polynomial. We used common factors and then factored a quadratic to reach our solution.