What happens when the divisor is a factor of the polynomial?
If the divisor is a factor, there will be no remainder.
When the divisor is a factor of the polynomial, it means that the polynomial can be divided evenly by the divisor with no remainder. In other words, the remainder of the polynomial division will be zero.
To check if the divisor is a factor of the polynomial, you can use the Remainder Theorem. According to the Remainder Theorem, if you divide a polynomial f(x) by x - a, the remainder will be equal to f(a). In other words, if you substitute the value of the root (a) into the polynomial and the result is zero, then the divisor (x - a) is a factor of the polynomial.
Here's an example to illustrate the process:
Let's say we have the polynomial f(x) = x^3 - 4x^2 + 5x - 2, and we want to check if x - 2 is a factor of this polynomial. We substitute the value of the root (a = 2) into the polynomial:
f(a) = f(2) = 2^3 - 4(2)^2 + 5(2) - 2
= 8 - 4(4) + 10 - 2
= 8 - 16 + 10 - 2
= 0
Since f(a) = 0, we conclude that x - 2 is a factor of the polynomial f(x).
In summary, when the divisor is a factor of the polynomial, the remainder of the polynomial division will be zero. To check if a divisor is a factor, we can use the Remainder Theorem by substituting the value of the root into the polynomial and checking if the result is zero.