Final answer:
When dividing a polynomial P(x) by a factor x - c results in a remainder of 0, c is a root or zero of P. This follows from the Factor Theorem, which states that a polynomial will have a root at c if and only if x - c is a factor of the polynomial.
Step-by-step explanation:
If we divide the polynomial P(x) by the factor x − c and we obtain a remainder of 0, then we know that c is a root or zero of P. This is because a remainder of 0 indicates that x − c exactly divides P(x), which implies that P(c) is equal to 0. According to the Factor Theorem, if x − c is a factor of P(x), then c is a root of the polynomial. The idea of a polynomial being factored is similar to simpler algebraic expressions, where if you divide an expression by one of its factors, the division is exact and the remainder is zero. For instance, if P(x) = x² - 5x + 6, and we divide it by x - 2 and get no remainder, it means that 2 is a root since P(2) = 2² - 5× 2 + 6 = 0.