A student uses polynomial division to divide $-3 x^4+15 x^3-x+5$ by $x-5$, and concludes that $x-5$ is not a factor of the polynomial.

The student's work is incorrect. What statements are true? Choose two correct answers.

[ ] The student should have used $k=5$, not $k=-5$.

[ ] The student should have used a placeholder zero for the $x^2$ term.

Question

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02-11-2026
Mathematics

Answered

A student uses polynomial division to divide $-3 x^4+15 x^3-x+5$ by $x-5$, and concludes that $x-5$ is not a factor of the polynomial.

The student's work is incorrect. What statements are true? Choose two correct answers.

[ ] The student should have used $k=5$, not $k=-5$.

[ ] The student should have used a placeholder zero for the $x^2$ term.

Verified Answer :

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GinnyAnswer GinnyAnswer · Answer 1 · 2026-11-02T01:51:08-05:00

Evaluate the polynomial using the Remainder Theorem: $P(5) = -3(5)^4 + 15(5)^3 - 5 + 5 = 0$, proving $x - 5$ is a factor.
Recognize that the student likely failed to use a zero placeholder for the missing $x^2$ term, which leads to incorrect synthetic division.
The two true statements are: the student should have used a placeholder zero for the $x^2$ term and the student should have correctly found that the remainder is 0, implying $x-5$ is a factor.

Explanation

Analyzing the Polynomial and Factor Theorem

We are given the polynomial $P(x) = -3x^4 + 15x^3 - x + 5$ and a divisor $D(x) = x - 5$. We want to determine if $D(x)$ is a factor of $P(x)$. According to the Remainder Theorem, $x - k$ is a factor of $P(x)$ if and only if $P(k) = 0$. In this case, $k = 5$. Evaluating the polynomial at $x = 5$:
\[P(5) = -3(5)^4 + 15(5)^3 - 5 + 5\]
\[P(5) = -3(625) + 15(125) - 0\]
\[P(5) = -1875 + 1875 = 0\]
Since $P(5) = 0$, $x - 5$ is indeed a factor of $P(x)$. The student's conclusion that it is not a factor is incorrect.

Identifying Errors in Synthetic Division

When performing synthetic division to divide a polynomial by $x - k$, we must include all powers of $x$ in descending order. If a power is missing (its coefficient is zero), we must include a zero as a placeholder in the synthetic division array. The polynomial $P(x) = -3x^4 + 15x^3 - x + 5$ has an $x^4$, $x^3$, $x^1$, and $x^0$ term. It is missing the $x^2$ term. Therefore, the coefficients to be used in synthetic division are $-3$, $15$, $0$ (the placeholder), $-1$, and $5$. Failing to include this zero will lead to incorrect division results.

Conclusion of Correct Statements

Given the analysis, we can identify two true statements that explain why the student's work was incorrect:

The student should have used a placeholder zero for the $x^2$ term in the synthetic division process.
Because $P(5) = 0$, the student should have correctly identified that $x - 5$ is a factor of the polynomial, rather than concluding it is not.

If the student failed to use the zero placeholder, they likely obtained a non-zero remainder, leading them to believe the factor was invalid.

Examples

In computer science and data processing, imagine you are storing coefficients of a polynomial in an array to perform calculations. If you miss a term, such as the $x^2$ term, and treat your array as [-3, 15, -1, 5] instead of [-3, 15, 0, -1, 5], your code will shift all subsequent operations incorrectly, resulting in wrong outputs. This problem highlights the necessity of maintaining data integrity, whether in math or programming, by ensuring every part of a system—or every term of a polynomial—is represented, even if it is just a placeholder value.