- Use the Polynomial Remainder Theorem to find the remainder of \(P(x)\) divided by \((x - a)\) as \(P(a)\).
- Set the divisor \(x + 1 = 0\) to find the evaluation point \(x = -1\).
- Substitute \(x = -1\) into \(P(x) = 5x^{52} - 4x^{10} - 11x^2 + 12\).
- Calculate the remainder: \(P(-1) = 5(1) - 4(1) - 11(1) + 12 = \boxed{2}\).
- Jessica evaluates the expression at \(x = \boxed{-1}\) and the remainder is \(\boxed{2}\).
Explanation
- Applying the Polynomial Remainder Theorem
The problem asks to find the remainder of the division of the polynomial \(P(x) = 5x^{52} - 4x^{10} - 11x^2 + 12\) by the linear divisor \(D(x) = x + 1\) using the Polynomial Remainder Theorem. The Polynomial Remainder Theorem states that the remainder of the division of a polynomial \(P(x)\) by a linear factor \((x - a)\) is simply the value of the polynomial evaluated at \(x = a\), i.e., \(P(a)\).
- Identifying the Evaluation Value
To apply the theorem, we must express the divisor \(x + 1\) in the form \((x - a)\). Comparing \(x + 1\) with \((x - a)\), we find that \(-a = 1\), which gives \(a = -1\). Therefore, Jessica must evaluate the polynomial \(P(x)\) at \(x = -1\) to find the remainder.
- Evaluating the Remainder
Now, we substitute \(x = -1\) into the polynomial \(P(x) = 5x^{52} - 4x^{10} - 11x^2 + 12\):
\[P(-1) = 5(-1)^{52} - 4(-1)^{10} - 11(-1)^2 + 12.\]
Since the exponents \(52, 10,\) and \(2\) are all even, \((-1)^{\text{even}} = 1\). Thus, the expression becomes:
\[P(-1) = 5(1) - 4(1) - 11(1) + 12.\]
Performing the arithmetic:
\[P(-1) = 5 - 4 - 11 + 12 = 1 - 11 + 12 = -10 + 12 = 2.\]
The remainder of the division is \(2\).
Examples
The Polynomial Remainder Theorem is a powerful tool in algebra that simplifies finding remainders of polynomial divisions without performing full polynomial long division. In computer science and error detection, evaluating polynomials at specific points can be used in checksum algorithms and error-correcting codes, such as Reed-Solomon codes, which protect data transmission by treating messages as coefficients of polynomials. Understanding this theorem helps in quickly assessing if a polynomial has a root (if the remainder is zero) or determining the value of a function efficiently.