Question

Factor. $$20 x^{5}-15 x^{3}-25 x$$

Short answer

Answer: The factored form of the expression $$20x^5 - 15x^3 - 25x$$ is $$5x(4x^4 - 3x^2 - 5)$$.

Step-by-step solution

1. Identify the GCF of the coefficients and the variables

First, let's find the GCF of the coefficients (20, -15, and -25) which is 5. Now, let's find the GCF of the variables which is \(x\) (as it is the lowest power of x in our expression).

2. Factor out the GCF

Now that we have identified the GCF which is 5x, we can factor it out from the given expression. To do this, we will apply the distributive property by dividing each term of the expression by the GCF: $$20x^5 - 15x^3 - 25x = 5x(4x^4 - 3x^2 - 5)$$

3. Check the factored expression

Now that we have factored out the GCF, let's check if the factored expression is correct by multiplying the GCF with the expression in parentheses: $$5x(4x^4 - 3x^2 - 5) = 5x * 4x^4 - 5x * 3x^2 - 5x * 5 = 20x^5 - 15x^3 - 25x$$ Since we got the original expression, our factoring is correct. The factored form of $$20x^5 - 15x^3 - 25x$$ is $$5x(4x^4 - 3x^2 - 5)$$.

Key concepts

Understanding the Greatest Common Factor (GCF)

A foundational step in factoring polynomials is to first find the Greatest Common Factor (GCF). The GCF is the largest factor that divides each term in a polynomial. It simplifies expressions and makes factoring more manageable.

To find the GCF of a polynomial:

• Identify the GCF of the coefficients. For instance, in the expression \(20x^5 - 15x^3 - 25x\), the coefficients are 20, -15, and -25. Calculate the GCF of these numbers, which is 5.
• Determine the GCF of the variables involved. The lowest power of the variable \(x\) in the expression is \(x^1\) because each term has at least one \(x\). Thus, the GCF for the variable part is \(x\).

Combine these to get the overall GCF for the expression, which is \(5x\). Recognizing the GCF is a critical step as it lays the groundwork for effectively simplifying the polynomial.

Applying the Distributive Property

The distributive property is a valuable tool in algebra, especially useful for factoring expressions. It allows us to express a polynomial in a factored form by 'distributing' common factors across terms. Let's explore how this works in our polynomial.

Once the GCF is identified, use it to factor out the common component from each term. For our example, we use the distributive property to separate the polynomial \(20x^5 - 15x^3 - 25x\) using the GCF of \(5x\). This process involves dividing each term in the polynomial by the GCF:

• The term \(20x^5\) becomes \(4x^4\) when divided by \(5x\).
• The term \(-15x^3\) becomes \(-3x^2\).
• The term \(-25x\) becomes \(-5\).

Putting these together, our expression is rewritten as \(5x(4x^4 - 3x^2 - 5)\). The distributive property ensures that when you multiply the GCF back through the expression in parentheses, you return to the original polynomial.

Mastering Factoring Expressions

Factoring expressions is a key algebraic skill that involves breaking down a polynomial into multiplicative components. This helps simplify expressions and solve polynomial equations efficiently.
Here's how it applies to our example:

• Start by determining the GCF (as discussed earlier), which informs how to structure the polynomial.
• Using the GCF, factor it out of the polynomial. In our case, the expression \(20x^5 - 15x^3 - 25x\) is transformed into the factored form \(5x(4x^4 - 3x^2 - 5)\).
• Check your work by expanding the factored form back to the original polynomial. Multiply the outside GCF by each term inside the parentheses to ensure accuracy. In doing so, you verify that the factored expression correctly simplifies the original polynomial, as observed in our example.

Mastering these steps allows you to handle even more complex polynomials confidently and recognize patterns in expressions that can be simplified. Through practice, factoring becomes an intuitive process, making algebra problems easier to tackle.