Verifying Factors Using The Remainder Theorem (x+5) And F(x)

Verifying factors of polynomials can sometimes feel like a puzzle, but powerful tools like the Remainder Theorem can make the process significantly easier. In this article, we'll explore how to use the Remainder Theorem to determine if $(x+5)$ is a factor of the function $f(x) = x^3 + 3x^2 - 25x - 75$. We'll break down the steps, explain the underlying concepts, and provide a clear, step-by-step guide to understanding this important theorem.

Understanding the Remainder Theorem

At its core, the Remainder Theorem provides a direct link between polynomial division and the value of the polynomial at a specific point. It states that if you divide a polynomial $f(x)$ by $(x - c)$, the remainder is equal to $f(c)$. This seemingly simple statement has profound implications for factor identification.

Let's delve deeper into the Remainder Theorem. Imagine you have a polynomial $f(x)$ and you want to divide it by a linear expression $(x - c)$. When you perform this division, you'll get a quotient, let's call it $q(x)$, and a remainder, which we'll denote as $r$. We can express this relationship mathematically as:

$f(x) = (x - c)q(x) + r$

Now, here's the crucial insight: what happens if we substitute $x = c$ into this equation?

$f(c) = (c - c)q(c) + r$

Since $(c - c)$ is zero, the entire first term on the right-hand side vanishes, leaving us with:

$f(c) = r$

This is the Remainder Theorem in action! It tells us that the value of the polynomial $f(x)$ when $x = c$ is exactly equal to the remainder when $f(x)$ is divided by $(x - c)$.

The Connection to Factors

How does this help us identify factors? Recall that a factor of a polynomial divides the polynomial evenly, leaving no remainder. In other words, if $(x - c)$ is a factor of $f(x)$, then dividing $f(x)$ by $(x - c)$ will result in a remainder of zero.

Combining this with the Remainder Theorem, we arrive at a powerful conclusion: $(x - c)$ is a factor of $f(x)$ if and only if $f(c) = 0$. This provides a straightforward method for checking if a linear expression is a factor of a given polynomial.

To solidify our understanding, let's consider a numerical example. Suppose we have the polynomial $f(x) = x^2 - 5x + 6$ and we want to check if $(x - 2)$ is a factor. According to the Remainder Theorem, we need to evaluate $f(2)$:

$f(2) = (2)^2 - 5(2) + 6 = 4 - 10 + 6 = 0$

Since $f(2) = 0$, the Remainder Theorem tells us that the remainder when $f(x)$ is divided by $(x - 2)$ is zero. Therefore, $(x - 2)$ is indeed a factor of $f(x)$. You can easily verify this by factoring the polynomial: $x^2 - 5x + 6 = (x - 2)(x - 3)$.

In contrast, let's check if $(x - 1)$ is a factor of the same polynomial. We evaluate $f(1)$:

$f(1) = (1)^2 - 5(1) + 6 = 1 - 5 + 6 = 2$

Since $f(1) = 2$, which is not zero, the Remainder Theorem tells us that $(x - 1)$ is not a factor of $f(x)$.

In summary, the Remainder Theorem offers a shortcut for determining factors of polynomials. Instead of performing long division, we simply evaluate the polynomial at a specific value. If the result is zero, we've found a factor. This theorem is a valuable tool in algebra and polynomial manipulation, making factor identification a more efficient process.

Applying the Remainder Theorem to Our Problem

Now, let's apply the Remainder Theorem to the specific problem at hand: determining if $(x + 5)$ is a factor of $f(x) = x^3 + 3x^2 - 25x - 75$.

First, we need to express $(x + 5)$ in the form $(x - c)$. Notice that $(x + 5)$ can be rewritten as $(x - (-5))$. This means that in our case, $c = -5$. According to the Remainder Theorem, if $(x + 5)$ is a factor of $f(x)$, then $f(-5)$ must equal zero.

Next, we evaluate $f(-5)$ by substituting $-5$ for $x$ in the polynomial:

$f(-5) = (-5)^3 + 3(-5)^2 - 25(-5) - 75$

Let's break down this calculation step by step:

• $(-5)^3 = -125$
• $3(-5)^2 = 3(25) = 75$
• $-25(-5) = 125$

Now, substitute these values back into the expression for $f(-5)$:

$f(-5) = -125 + 75 + 125 - 75$

Combining the terms, we get:

$f(-5) = 0$

The result is zero! This is a crucial finding. Since $f(-5) = 0$, the Remainder Theorem confirms that $(x + 5)$ is indeed a factor of $f(x) = x^3 + 3x^2 - 25x - 75$.

Therefore, by applying the Remainder Theorem, we have efficiently verified that $(x + 5)$ is a factor of the given polynomial. This method saves us the effort of performing polynomial long division, which can be more time-consuming and prone to errors.

Alternative Verification Methods

While the Remainder Theorem provides a swift and elegant solution, it's worth mentioning that there are alternative methods to verify if $(x + 5)$ is a factor of $f(x)$. One such method is polynomial long division. If we were to divide $f(x) = x^3 + 3x^2 - 25x - 75$ by $(x + 5)$ using long division, we would find that the remainder is zero, thus confirming that $(x + 5)$ is a factor.

Another approach is to attempt to factor the polynomial $f(x)$ directly. By factoring, we hope to express $f(x)$ as a product of linear factors, one of which might be $(x + 5)$. In this case, we can factor $f(x)$ as follows:

$f(x) = x^3 + 3x^2 - 25x - 75$

First, we can try factoring by grouping:

$f(x) = (x^3 + 3x^2) + (-25x - 75)$

Factor out the greatest common factor from each group:

$f(x) = x^2(x + 3) - 25(x + 3)$

Now, we can factor out the common binomial factor $(x + 3)$:

$f(x) = (x + 3)(x^2 - 25)$

Notice that $(x^2 - 25)$ is a difference of squares, which can be further factored as $(x + 5)(x - 5)$. Therefore,

$f(x) = (x + 3)(x + 5)(x - 5)$

From this factored form, it's clear that $(x + 5)$ is indeed a factor of $f(x)$.

In conclusion, while the Remainder Theorem offers a direct and efficient method for verifying factors, other techniques like polynomial long division and factoring can also be employed. The choice of method often depends on personal preference and the specific characteristics of the polynomial in question. However, understanding the Remainder Theorem provides a powerful tool in your algebraic arsenal.

Step-by-Step Breakdown

Let's summarize the steps we took to verify if $(x + 5)$ is a factor of $f(x) = x^3 + 3x^2 - 25x - 75$ using the Remainder Theorem:

1. Identify the value of 'c': Rewrite the factor $(x + 5)$ in the form $(x - c)$. In this case, $(x + 5) = (x - (-5))$, so $c = -5$.
2. Evaluate f(c): Substitute the value of $c$ (which is -5) into the function $f(x)$: $f(-5) = (-5)^3 + 3(-5)^2 - 25(-5) - 75$
3. Simplify the expression: Calculate the value of $f(-5)$: $f(-5) = -125 + 3(25) + 125 - 75$ $f(-5) = -125 + 75 + 125 - 75$ $f(-5) = 0$
4. Interpret the result: If $f(c) = 0$, then $(x - c)$ is a factor of $f(x)$. Since $f(-5) = 0$, we conclude that $(x + 5)$ is a factor of $f(x)$.

By following these steps, you can confidently apply the Remainder Theorem to verify factors of polynomials. This method provides a valuable shortcut and a deeper understanding of the relationship between polynomial division and function evaluation.

Conclusion

The Remainder Theorem is a powerful tool for determining if a linear expression is a factor of a polynomial. By evaluating the polynomial at a specific value and checking if the result is zero, we can efficiently verify factors without resorting to long division. In the case of $f(x) = x^3 + 3x^2 - 25x - 75$ and the potential factor $(x + 5)$, the Remainder Theorem provided a clear and concise way to confirm that $(x + 5)$ is indeed a factor. Understanding and applying the Remainder Theorem enhances your ability to work with polynomials and solve algebraic problems more effectively.