Video: SAT Practice Test 1 • Section 3 • Question 5

Consider the equation 𝑎𝑥⁴ + 𝑏𝑥³ + 𝑐𝑥² = 0, where 𝑎, 𝑏, and 𝑐 are constants. If the equation has roots 2, −3, and 0, which of the following is a factor of 𝑎𝑥⁴ + 𝑏𝑥³ + 𝑐𝑥² = 0? [A] 𝑥 + 3 [B] 𝑥 − 4 [C] 𝑥 + 2 [D] 𝑥 − 3

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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.

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