Select the correct answer from each drop-down menu.

Use the remainder theorem to verify this statement: $(x+5)$ is a factor of the function $f(x)=x^3+3x^2-25x-75$.

Find the [ Choose an answer ] of $f(x)$ and $x+5$.
The [ Choose an answer ] of this operation is 0.
Therefore, $(x+5)$ is [ Choose an answer ] of function $f$.
So $f($ [ Choose an answer ] $) = 0$.

Question

04-11-2026
Mathematics

Answered

Select the correct answer from each drop-down menu.

Use the remainder theorem to verify this statement: $(x+5)$ is a factor of the function $f(x)=x^3+3x^2-25x-75$.

Find the [ Choose an answer ] of $f(x)$ and $x+5$.
The [ Choose an answer ] of this operation is 0.
Therefore, $(x+5)$ is [ Choose an answer ] of function $f$.
So $f($ [ Choose an answer ] $) = 0$.

Verified Answer :

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GinnyAnswer GinnyAnswer · Answer 1 · 2026-11-04T01:45:38-05:00

The Remainder Theorem dictates that dividing $f(x)$ by $(x-c)$ results in a remainder of $f(c)$.
Evaluating the polynomial at $c = -5$ gives $f(-5) = (-5)^3 + 3(-5)^2 - 25(-5) - 75 = 0$.
Since the remainder is $0$, $(x+5)$ is confirmed to be a factor of the function $f(x)$.
Final statements: 1. remainder, 2. remainder, 3. a factor, 4. -5. The final result is that the factor is verified as $\boxed{f(-5)=0}$.

Explanation

Analysis of the Problem

The goal is to verify that $(x+5)$ is a factor of the polynomial $f(x) = x^3 + 3x^2 - 25x - 75$ using the Remainder Theorem. The Remainder Theorem states that the remainder of the division of a polynomial $f(x)$ by a linear divisor of the form $(x - c)$ is equal to $f(c)$. For the divisor $(x + 5)$, we can write this as $(x - (-5))$, which identifies $c = -5$. If $(x+5)$ is a factor of $f(x)$, the remainder obtained upon division must be zero, implying $f(-5) = 0$.

Applying the Remainder Theorem

To find the remainder of the division of $f(x) = x^3 + 3x^2 - 25x - 75$ by $(x+5)$, we evaluate the function at $x = -5$.
Calculating $f(-5)$:
\[f(-5) = (-5)^3 + 3(-5)^2 - 25(-5) - 75\]
\[f(-5) = -125 + 3(25) + 125 - 75\]
\[f(-5) = -125 + 75 + 125 - 75\]
\[f(-5) = 0\]
Since the evaluation of the function at $-5$ results in $0$, the remainder of the division is $0$.

Completing the Verification

Based on the result $f(-5) = 0$, we can fill in the required blanks:

Find the remainder of $f(x)$ and $(x+5)$.
The remainder of this operation is $0$.
Therefore, $(x+5)$ is a factor of function $f$.
So $f($ -5 $) = 0$.

Thus, we have successfully verified that $(x+5)$ is a factor of the given polynomial.

Examples

The Remainder Theorem is a fundamental tool in algebra used to check if a linear expression is a factor of a polynomial. For instance, in computer science and data encryption, polynomials are often used to represent data or keys. Being able to quickly determine if one polynomial divides another without performing the full division (like long division) saves significant computational resources. Understanding this principle allows for efficient error detection and data verification in various digital systems.