### Video Transcript

Consider the equation 𝑎𝑥 to the
fourth power plus 𝑏𝑥 cubed plus 𝑐𝑥 squared equals zero, where 𝑎, 𝑏, and 𝑐 are
constants. If the equation has roots two,
negative three, and zero, which of the following is a factor of 𝑎𝑥 to the fourth
power plus 𝑏𝑥 cubed plus 𝑐𝑥 squared equals zero? Is it option A) 𝑥 plus three,
option B) 𝑥 minus four, option C) 𝑥 plus two, or option D) 𝑥 minus three?

The factor theorem for any
polynomial states that if 𝑓 of 𝑎 equals zero, then 𝑥 minus 𝑎 is a factor. In our question, we are told that
the equation has roots two, negative three, and zero. This means that if we let 𝑓 of 𝑥
equal 𝑎𝑥 to the fourth power plus 𝑏𝑥 cubed plus 𝑐𝑥 squared, then 𝑓 of two
equals zero, 𝑓 of negative three equals zero, and 𝑓 of zero also equals zero.

As 𝑓 of two is equal to zero, 𝑥
minus two is a factor. As 𝑓 of minus three equals zero,
𝑥 plus three is a factor. Finally, as 𝑓 of zero equals zero,
then 𝑥 minus zero is a factor. As 𝑥 minus zero is equal to 𝑥, we
can say that 𝑥 is a factor. We have, therefore, found three
factors of the equation: 𝑥 minus two, 𝑥 plus three, and 𝑥. The only one of these that matches
our four options that we were given is 𝑥 plus three.

We can, therefore, say that the
equation 𝑎𝑥 to the fourth power plus 𝑏𝑥 cubed plus 𝑐𝑥 squared equals zero with
roots two, negative three, and zero has a factor of 𝑥 plus three.