- Identify the divisor \(x+1\) as \(x - (-1)\), requiring the evaluation of the polynomial at \(x = -1\).
- Substitute \(x = -1\) into \(P(x) = 5x^{52}-4x^{15}-11x^2+12\).
- Calculate the powers and simplify the resulting expression: \(5(1) - 4(-1) - 11(1) + 12 = 10\).
- Conclude that the remainder is \(\boxed{10}\).
Explanation
- Understanding the Remainder Theorem
To find the remainder of the polynomial division \(\frac{5x^{52}-4x^{15}-11x^2+12}{x+1}\) using the remainder theorem, we identify the polynomial as \(P(x) = 5x^{52}-4x^{15}-11x^2+12\). The remainder theorem states that the remainder of a polynomial \(P(x)\) when divided by a linear factor \(x - c\) is equal to \(P(c)\). In this case, the divisor is \(x + 1\), which can be written as \(x - (-1)\), meaning \(c = -1\). Therefore, Jessica needs to evaluate the polynomial at \(x = -1\).
- Calculating the Remainder
Now, we substitute \(x = -1\) into the polynomial \(P(x) = 5x^{52}-4x^{15}-11x^2+12\):
\[P(-1) = 5(-1)^{52} - 4(-1)^{15} - 11(-1)^2 + 12\]
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:
\[P(-1) = 5(1) - 4(-1) - 11(1) + 12\]
\[P(-1) = 5 + 4 - 11 + 12\]
\[P(-1) = 9 - 11 + 12 = 10\]
Thus, the remainder is 10.
- Conclusion
Jessica evaluates the numerator of the rational expression when \(x = -1\). She concludes that the remainder of the division is \(\boxed{10}\).
Examples
The remainder theorem is a powerful tool in algebra used to simplify polynomial evaluations. For example, if you are designing a computer graphics algorithm that involves complex polynomial paths for a character's movement, you might need to check if a path (polynomial) passes through a certain point. Instead of performing full polynomial division, which is computationally expensive, the remainder theorem allows you to quickly evaluate the polynomial at a specific point, saving time and resources. This principle is fundamental in computer science, cryptography, and engineering.