\begin{tabular}{|l|}\begin{tabular}{|l} 7. Please calculate step by step \\ the remainder that results from the \\ quotient expressed below using \\ the Polynomial Remainder \\ Theorem. \\ \( \left(-5 x^{3}+x^{2}-4 x\right) \div(x-3) \) \\ Please be explicit in writing each \\ step and number each step. Also, \\ please write a brief explanation \\ with each step to explain what you \\ are doing. \\ Please conclude with a statement \\ that explains whether \( (x-3) \) is a \\ factor of the stated polynomial. \end{tabular} \\ \hline\end{tabular}

To find the remainder of the polynomial \( -5x^{3} + x^{2} - 4x \) when divided by \( (x-3) \) using the Polynomial Remainder Theorem, follow these steps: **Step 1:** Identify the polynomial and divisor. We start with the polynomial \( P(x) = -5x^{3} + x^{2} - 4x \) and the divisor \( x - 3 \). We need to evaluate \( P(3) \) to find the remainder. **Step 2:** Substitute \( x = 3 \) into the polynomial. Compute \( P(3) \): \[ P(3) = -5(3)^{3} + (3)^{2} - 4(3) \] Calculate \( (3)^{3} = 27 \), \( (3)^{2} = 9 \), and \( -4(3) = -12 \). **Step 3:** Substitute and calculate step by step. Now plug in these values: \[ P(3) = -5(27) + 9 - 12 \] Calculate \( -5(27) = -135 \): \[ P(3) = -135 + 9 - 12 \] **Step 4:** Perform the final addition. Now, compute \( -135 + 9 = -126 \) and then \( -126 - 12 = -138 \): \[ P(3) = -138 \] **Conclusion:** The remainder when \( -5x^{3} + x^{2} - 4x \) is divided by \( x - 3 \) is \( -138 \). Since the remainder is not zero, \( (x - 3) \) is not a factor of the polynomial \( -5x^{3} + x^{2} - 4x \). --- In history, the Polynomial Remainder Theorem dates back to ancient Greece, with mathematicians like Euclid exploring the properties of polynomial equations. This theorem has evolved over time, shaping modern algebra and calculus. Understanding its root helps anchor our knowledge in mathematical history, showcasing how ancient concepts continue to play a vital role in today's problem-solving techniques! In real-world applications, the Polynomial Remainder Theorem simplifies complex problems, like finding roots in engineering or computer science. When dealing with polynomial models, engineers can predict system behaviors by determining specific outcomes (like the time to failure of a system) without performing long division, making their analyses much more efficient and effective!