Question

Let the function P be defined by P(x) = x³ +7x² - 26x - 72 where (x+9) is a factor. To rewrite the function as the product of two factors, long division was used but an error was made: How can we tell by looking at the remainder that an error was made somewhere

Answer 1

By looking at the remainder in the long division process, we can tell that an error was made if the remainder is non-zero. In this case, a non-zero remainder would indicate that the divisor (x + 9) is not a factor of the polynomial P(x), contrary to the assumption.

Step-by-step explanation:

When performing long division to factorize a polynomial, the remainder at the end of the division process should ideally be zero for the divisor to be considered a factor. In this scenario, the divisor is (x + 9). If the remainder is non-zero, it means that the polynomial cannot be completely factored by (x + 9), and an error has likely occurred in the factorization process.

To elaborate, the division process involves dividing the polynomial P(x) by (x + 9). If (x + 9) is indeed a factor, the division should result in a quotient without any remainder. A non-zero remainder indicates that there are additional terms not accounted for, suggesting that the factorization process was not executed correctly.

In summary, if the remainder is not zero in the long division of P(x) by (x + 9), it signifies an error in the assumption that (x + 9) is a factor of the polynomial. The correct factorization should leave no remainder, confirming the validity of the divisor as a factor of the given polynomial.