- Identify the leading term of the dividend as \(3x^4\).
- Identify the leading term of the divisor as \(x^2\).
- Calculate the first quotient term by dividing the leading term of the dividend by the leading term of the divisor: \(\frac{3x^4}{x^2} = 3x^2\).
- The divisor term is \(\boxed{x^2}\).
Explanation
- Analysis of the Polynomial Division
To determine what term the leading term of the dividend must be divided by to obtain the leading term of the quotient, we perform polynomial division of the polynomials \(3x^4 + 7x^3 + 2x^2 + 13x + 5\) and \(x^2 + 3x + 1\).
In polynomial long division, the first term of the quotient is found by dividing the term with the highest exponent of the dividend by the term with the highest exponent of the divisor.
The leading term of the dividend is \(3x^4\).
The leading term of the divisor is \(x^2\).
Therefore, the first term of the quotient is:
\[\frac{3x^4}{x^2} = 3x^2\]
This matches the provided information in the problem. The question asks for the term that \(3x^4\) is divided by to get the quotient term \(3x^2\). This is simply the leading term of the divisor.
- Determining the divisor term
The division process is as follows:
We divide the leading term of the dividend, \(3x^4\), by the leading term of the divisor, \(x^2\).
\[\frac{3x^4}{x^2} = 3x^2\]
This confirms that the term we are dividing by is the leading term of the divisor, which is \(x^2\).
Thus, the term \(3x^2\) is the result of dividing \(3x^4\) by \(x^2\).
- Conclusion
The calculation \(\frac{3x^4}{x^2} = 3x^2\) confirms that the leading term of the divisor, \(x^2\), is the divisor term that produces the quotient term \(3x^2\). The final answer is \(x^2\).
Examples
Polynomial long division is a fundamental tool in algebra used for simplifying rational expressions and finding roots of polynomials. For example, in engineering, when designing control systems, one often needs to perform long division of transfer functions to simplify the analysis of system stability and response time. Mastering this process ensures accuracy when solving complex algebraic equations.