In a polynomial division problem, the dividend does not have an $x^2$-term. When the problem is written in long division form, what should be the coefficient of the $x^2$-term in the dividend?
A. -1
B. 1
C. 2
D. 0
Answer (1)
When a polynomial does not have an x 2 term, it means the coefficient of x 2 is zero.
In long division, we include 0 x 2 to maintain place value.
The coefficient of the x 2 term in the dividend is 0.
Therefore, the answer is 0 .
Explanation
Understanding the Missing Term In polynomial division, the dividend is the polynomial being divided. If the dividend does not have an x 2 term, it means that the coefficient of the x 2 term is zero. When writing the polynomial in long division form, we explicitly include the x 2 term with a coefficient of 0 to maintain proper place value for the other terms.
Illustrative Example For example, if we are dividing x 3 + 5 x + 6 by some polynomial, we would write the dividend as x 3 + 0 x 2 + 5 x + 6 in long division form. This ensures that the terms are aligned correctly during the division process.
Conclusion Therefore, the coefficient of the x 2 term in the dividend should be 0.
Examples
Polynomial division is used in various applications, such as simplifying rational expressions and solving polynomial equations. For example, when designing a bridge, engineers might use polynomial division to analyze the load distribution and ensure the structure's stability. By representing the load and structural properties as polynomials, engineers can use polynomial division to simplify the analysis and find critical points where the structure might be more susceptible to stress. This helps in optimizing the design and ensuring the bridge's safety and durability.
