Grades 11-12 Video Solutions 2023-2023_11-12

Video Transcription

Problem number 25. Part of the 5th degree polynomial shown to the right cannot be seen because of an inkblot. It is known that all 5 roots of the polynomial are integers. What  is the highest power of x-1 that divides the polynomial? Okay, whenever I see something with integer roots and integer coefficients in a polynomial,  I immediately think of Vietta's formulas. If you don't remember Vietta's formulas, this is a good time to pause the video and go learn about them.  Since a constant term is equal to negative 7, the product of the roots must be 7 according  to Vietta's formulas. With 5 roots, what we must have if they're all integers is that there must  be one root of plus or minus 7 and the other 4 roots will be equal to plus or minus 1. We have to look at the signs to make sure that the final signs all work out correctly.  The coefficient of x to the 4 is negative 11. That tells us that the sum of the roots is 11.  And so the only way to get a sum of 11 with the roots we have, so the 7s and the 1s, is 1 plus 1  plus 1 plus 1 plus 7. So all the roots must be positive. But that makes our life easy. We can  factor this polynomial as x minus 1 to the power of 4 times x minus 7. And so our correct answer is d and we're done.

Video Summary

The video discusses a problem involving a 5th degree polynomial with integer roots, partially obscured by an inkblot. Using Vieta's formulas, it concludes that the constant term implies the roots' product is 7. Thus, one root must be ±7, and the others ±1. Given the sum of roots is 11 (from the x⁴ coefficient), all roots must be positive: four 1s and one 7. This allows factoring of the polynomial as \((x - 1)^4(x - 7)\). Hence, the highest power of \(x - 1\) dividing the polynomial is 4. The answer is "d".

Keywords

5th degree polynomial

integer roots

Vieta's formulas

factoring polynomial

roots product