Given that 𝑥 minus 𝑎 is a factor of 𝑓 of 𝑥, what is the remainder when 𝑓 of 𝑥 is divided by 𝑥 minus 𝑎?
The factor theorem states that the binomial 𝑥 minus 𝑎 is a factor of the polynomial 𝑓 of 𝑥 if, and only if, 𝑓 of 𝑎 is zero. So if we think of factors, we multiply factors together to equal our polynomial. So let’s simplify this, we can think about it. If we would multiply two and three together, they equal six. So our factors are two and three. Those would be our 𝑥 minus 𝑎 examples, and then six would be our polynomial 𝑓 of 𝑥. So if we would take our polynomial, which we are representing by six, and dividing it by one of the factors, two or three, what would be our remainder?
We shouldn’t get a remainder because if it’s a factor, that means it goes in there perfectly. So our remainder would be zero.