- Use the Remainder Theorem, which states the remainder of \(P(x) / (x - a)\) is \(P(a)\).
- Identify that for the divisor \(x + 1\), we have \(a = -1\).
- Evaluate the polynomial \(P(-1) = 5(-1)^{52} - 4(-1)^{15} - 11(-1)^2 + 12\).
- Simplify the expression to find the remainder: \(P(-1) = 10\), so the remainder is \(\boxed{10}\).
Explanation
- Understanding the Remainder Theorem
To find the remainder of the division of a polynomial \(P(x)\) by a linear expression of the form \((x - a)\), we can use the Remainder Theorem. The theorem states that the remainder of the division \(\frac{P(x)}{x - a}\) is equal to \(P(a)\). In this problem, the polynomial is \(P(x) = 5x^{52} - 4x^{15} - 11x^2 + 12\) and the divisor is \(x + 1\). Since \(x + 1 = x - (-1)\), we have \(a = -1\). Therefore, to find the remainder, we need to evaluate the polynomial at \(x = -1\).
- Evaluating the Polynomial at x = -1
We substitute \(x = -1\) into the expression \(P(x) = 5x^{52} - 4x^{15} - 11x^2 + 12\):
\[P(-1) = 5(-1)^{52} - 4(-1)^{15} - 11(-1)^2 + 12\]
We then evaluate the powers of \(-1\):
- Since 52 is an even number, \((-1)^{52} = 1\).
- Since 15 is an odd number, \((-1)^{15} = -1\).
- Since 2 is an even number, \((-1)^2 = 1\).
Substituting these values back into the expression:
\[P(-1) = 5(1) - 4(-1) - 11(1) + 12\]
\[P(-1) = 5 + 4 - 11 + 12\]
\[P(-1) = 9 - 11 + 12 = -2 + 12 = 10\]
Thus, the remainder of the division is 10.
- Conclusion
Jessica evaluates the numerator of the rational expression when \(x = -1\). She concludes that the remainder of the division is 10. The completed statements are:
- Jessica evaluates the numerator of the rational expression when \(x = -1\).
- She concludes that the remainder of the division is 10.
The final remainder is \(\boxed{10}\).
Examples
The Remainder Theorem is highly useful in computer science and algebra when you need to quickly check if one polynomial is a factor of another. For instance, if you are designing a digital communication system that uses cyclic redundancy checks (CRC) for error detection, you often work with polynomials over finite fields. By applying the Remainder Theorem, you can efficiently determine the "remainder" of a message polynomial divided by a generator polynomial, which helps identify if the data was transmitted correctly without corruption. Understanding this theorem allows engineers to perform these checks with simple substitutions rather than performing long, time-consuming polynomial division.