Factoring Polynomials: A Step-by-Step Guide

Dec 16, 2025 by Alex Johnson 44 views

Hey there, math enthusiasts! Today, we're diving deep into the world of algebra to tackle a common challenge: factoring polynomials. If you've ever stared at an expression like $2x^3 + 26x^2 + 80x$ and felt a little overwhelmed, don't worry – you're not alone! Factoring is a fundamental skill in mathematics, and mastering it can unlock a whole new level of understanding when it comes to solving equations, graphing functions, and much more. In this guide, we'll break down the process of completely factoring this specific expression, step by painstaking step. We'll not only show you how to do it but also why each step is important, ensuring you build a solid foundation. So, grab your notebooks, sharpen those pencils, and let's get ready to unravel the mystery of factoring polynomials!

Understanding the Goal: What Does "Completely Factor" Mean?

Before we jump into the nitty-gritty of factoring our expression $2x^3 + 26x^2 + 80x$ , let's make sure we're all on the same page about what "completely factor" actually signifies. In essence, factoring is like taking a complex number or expression and breaking it down into its simplest multiplicative building blocks, much like dismantling a LEGO set into individual bricks. When we talk about completely factoring a polynomial, we mean rewriting it as a product of irreducible factors. For polynomials with integer coefficients, like the one we're working with, this typically means factoring it into linear factors (factors with an 'x' term) and irreducible quadratic factors (quadratic factors that cannot be factored further using real numbers). The goal is to reach a point where none of the resulting factors can be broken down any further. Think of it as reaching prime numbers when factoring integers – you can't divide them any smaller. This process is crucial because it simplifies expressions, reveals the roots (or x-intercepts) of a polynomial equation, and is a cornerstone for many higher-level mathematical concepts. We'll be aiming to express $2x^3 + 26x^2 + 80x$ as a product of terms that, when multiplied back together, give us the original expression, and none of these terms can be factored further.

Step 1: Identifying the Greatest Common Factor (GCF)

Our journey to completely factor the expression $2x^3 + 26x^2 + 80x$ begins with the most fundamental step in polynomial factoring: identifying and extracting the Greatest Common Factor (GCF). The GCF is the largest factor that divides into every term of the polynomial. Finding the GCF is paramount because it simplifies the remaining polynomial, making subsequent factoring steps significantly easier. Let's examine our expression term by term. We have $2x^3$ , $26x^2$ , and $80x$ . First, let's look at the coefficients: 2, 26, and 80. To find the GCF of these numbers, we can list their prime factors. The prime factorization of 2 is simply 2. For 26, it's $2 \times 13$ . And for 80, it's $2 \times 2 \times 2 \times 2 \times 5$ , or $2^4 \times 5$ . Now, we look for the common prime factors shared by all three numbers. The only prime factor that appears in all three is 2. Therefore, the GCF of the coefficients 2, 26, and 80 is 2. Next, let's consider the variable parts: $x^3$ , $x^2$ , and $x$ . To find the GCF of the variable terms, we take the lowest power of each variable present in all terms. In this case, the lowest power of 'x' is $x^1$ , or simply 'x'. So, the GCF of $x^3$ , $x^2$ , and $x$ is $x$ . Combining the GCF of the coefficients and the GCF of the variable terms, we find that the overall GCF of the entire expression $2x^3 + 26x^2 + 80x$ is $2x$ . Now, we factor out this GCF from each term. To do this, we divide each term by $2x$ :

$(2x^3) / (2x) = x^2$
$(26x^2) / (2x) = 13x$
$(80x) / (2x) = 40$

So, after factoring out the GCF, our expression becomes $2x(x^2 + 13x + 40)$ . This is a significant simplification, and we've successfully completed the first crucial step in completely factoring the polynomial. Remember, always look for the GCF first – it's your golden ticket to easier factoring!

Step 2: Factoring the Remaining Quadratic Expression

Now that we've successfully extracted the GCF, our expression is reduced to $2x(x^2 + 13x + 40)$ . Our next objective in completely factoring $2x^3 + 26x^2 + 80x$ is to factor the quadratic expression that remains inside the parentheses: $x^2 + 13x + 40$ . This is a trinomial of the form $ax^2 + bx + c$ , where $a=1$ , $b=13$ , and $c=40$ . When $a=1$ , factoring a quadratic trinomial often involves finding two numbers that satisfy two conditions: their product must equal the constant term ( $c$ ), and their sum must equal the coefficient of the middle term ( $b$ ). In our case, we need to find two numbers that multiply to 40 and add up to 13. Let's list the pairs of factors for 40 and check their sums:

1 and 40: Sum = 41
2 and 20: Sum = 22
4 and 10: Sum = 14
5 and 8: Sum = 13

Aha! We've found our pair: 5 and 8. Their product is $5 \times 8 = 40$ , and their sum is $5 + 8 = 13$ . These are the numbers we need!

Since we've found these two numbers, we can now rewrite the middle term ( $13x$ ) as the sum of two terms using these numbers: $5x + 8x$ . So, our quadratic expression becomes $x^2 + 5x + 8x + 40$ . This technique is often called factoring by grouping, and it's a common method when $a=1$ . Now, we group the first two terms and the last two terms:

$(x^2 + 5x) + (8x + 40)$

Next, we find the GCF of each group.

For the first group, $(x^2 + 5x)$ , the GCF is $x$ . Factoring it out gives $x(x + 5)$ .
For the second group, $(8x + 40)$ , the GCF is 8. Factoring it out gives $8(x + 5)$ .

Now, our expression looks like this: $x(x + 5) + 8(x + 5)$ . Notice that both terms now share a common binomial factor: $(x + 5)$ . We can factor this common binomial out, leaving the remaining terms ( $x$ and $+8$ ) as the other factor.

So, factoring $x^2 + 13x + 40$ gives us $(x + 5)(x + 8)$ . This quadratic expression is now completely factored into two linear factors.

Step 3: Combining Factors for the Final Answer

We've reached the final stage in our quest to completely factor the expression $2x^3 + 26x^2 + 80x$ . In Step 1, we identified and factored out the Greatest Common Factor (GCF), which was $2x$ . This left us with the expression $2x(x^2 + 13x + 40)$ . Then, in Step 2, we successfully factored the quadratic trinomial $x^2 + 13x + 40$ into its two linear factors, $(x + 5)$ and $(x + 8)$ .

Now, all that's left to do is to put all the pieces back together. We simply combine the GCF we extracted with the factored form of the quadratic. Our original expression, when completely factored, will be the product of the GCF and the factored quadratic.

So, we take the GCF, $2x$ , and multiply it by the factored form of the quadratic, $(x + 5)(x + 8)$ .

This gives us our final, completely factored expression:

2x(x + 5)(x + 8)

To verify our answer, we can always multiply these factors back together.

First, multiply the binomials: $(x + 5)(x + 8) = x(x + 8) + 5(x + 8) = x^2 + 8x + 5x + 40 = x^2 + 13x + 40$ .

Now, multiply this result by the GCF, $2x$ : $2x(x^2 + 13x + 40) = 2x(x^2) + 2x(13x) + 2x(40) = 2x^3 + 26x^2 + 80x$ .

We've arrived back at our original expression, which confirms that our factoring is correct and complete! Each factor ( $2x$ , $x+5$ , and $x+8$ ) is irreducible, meaning it cannot be factored any further using real numbers.

Why is Factoring So Important?

Understanding how to completely factor expressions like $2x^3 + 26x^2 + 80x$ is more than just an algebraic exercise; it's a gateway to solving a wide array of mathematical problems. One of the most significant applications of factoring polynomials is in solving polynomial equations. If we set our factored expression equal to zero, $2x(x + 5)(x + 8) = 0$ , we can easily find the roots (or solutions) of the equation. By the Zero Product Property, if a product of factors equals zero, then at least one of the factors must be zero. Therefore, we can set each factor equal to zero and solve for $x$ :

$2x = 0 \implies x = 0$
$x + 5 = 0 \implies x = -5$
$x + 8 = 0 \implies x = -8$

These values, $x = 0, -5, -8$ , are the roots of the polynomial equation $2x^3 + 26x^2 + 80x = 0$ . Knowing these roots is crucial for graphing the polynomial, as they represent the x-intercepts of its curve. Furthermore, factoring is indispensable in simplifying rational expressions (fractions involving polynomials), performing polynomial division, and is a foundational concept for calculus and other advanced mathematical disciplines. Mastering factoring empowers you to manipulate and understand algebraic expressions with greater confidence and efficiency.

Conclusion: Your Factoring Skills Unleashed!

We've successfully navigated the process of completely factoring the expression $2x^3 + 26x^2 + 80x$ . We started by identifying and extracting the Greatest Common Factor ( $2x$ ), simplifying the expression to $2x(x^2 + 13x + 40)$ . Then, we focused on the quadratic trinomial, finding two numbers (5 and 8) that multiplied to 40 and added to 13, allowing us to factor it into $(x + 5)(x + 8)$ . Finally, we combined our GCF and the factored quadratic to arrive at the fully factored form: $2x(x + 5)(x + 8)$ . Remember, the key steps are always:

Find the GCF: Always start by looking for the greatest common factor among all terms.
Factor the remaining polynomial: This might involve factoring a quadratic trinomial, using grouping, or other techniques depending on the number and type of terms left.
Check your work: Multiply your factors back together to ensure you get the original expression.

Keep practicing these steps with different expressions, and you'll find your factoring skills rapidly improving. This fundamental algebraic skill will serve you well in many areas of mathematics. For further exploration into the fascinating world of polynomial manipulation and algebraic techniques, you might find the resources at Khan Academy to be incredibly helpful. Happy factoring!