50. Remainder and factor theorem

The Remainder Theorem and the Factor Theorem are two fundamental concepts in mathematics that are often required in the Enem test. Knowing and understanding these theorems can be the difference between solving a question correctly or not.

The Remainder Theorem is a theorem that allows us to find the remainder of a division of a polynomial P(x) by a binomial of the form x - a. According to the theorem, the remainder of this division is equal to the value of the polynomial P(a). That is, if we want to find the remainder of the division of a polynomial P(x) by x - a, just substitute x for a in the polynomial. For example, if we want to find the remainder of the division of the polynomial P(x) = x³ - 2x² + x - 3 by x - 1, just replace x by 1 in the polynomial P(x), that is, P(1) = 1³ - 2*1² + 1 - 3 = -3. So the remainder of division is -3.

The Factor Theorem, on the other hand, tells us that if a polynomial P(x) has a root a, then x - a is a factor of the polynomial. This means that if we substitute x for a in the polynomial and the result is zero, then x - a is a factor of the polynomial. For example, if we have the polynomial P(x) = x³ - 2x² + x - 3 and find that P(1) = 0, then x - 1 is a factor of the polynomial. We can confirm this by performing the division of the polynomial P(x) by x - 1. If the remainder of the division is zero, then x - 1 is indeed a factor of the polynomial.

These two theorems are closely related and are very useful in solving problems involving polynomials. They allow us to find the factors and roots of a polynomial, which is essential for solving polynomial equations.

In the ENEM exam, questions involving the Remainder Theorem and the Factor Theorem usually involve solving polynomial equations or factoring polynomials. These questions may seem complex at first glance, but with an understanding of these theorems, they become much more manageable.

For example, a question might ask to find the roots of a third degree polynomial. Using the Factor Theorem, we can start by guessing a root of the polynomial and then use that root to find a factor of the polynomial. Once we find a factor, we can divide the polynomial by that factor to get a lower degree polynomial. We can then repeat the process to find the other roots of the polynomial.

In summary, the Remainder Theorem and the Factor Theorem are powerful tools in mathematics that allow us to solve complex problems efficiently. They are fundamental for understanding polynomials and are often required in the Enem test. Therefore, it is essential that students understand and know how to apply these theorems.

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