Online Course Discussion Forum

Math Challenge II-A 4.25 and 4.29 Help

 
 
Picture of Zhi Feng Etan Kiang
Math Challenge II-A 4.25 and 4.29 Help
by Zhi Feng Etan Kiang - Saturday, 27 June 2020, 6:16 PM
 

How are you? I am stuck on 4.25 and 4.29. For 4.25, I am completely stumped, but for 4.29, I tried turning the equation into 1+a+a^2+a^3+a^4+a^5-a-a^2-a^3, but when I continued from there, the equation only got more complicated. Please help me on 4.25 and 4.29, and thank you!

 
Picture of Dr. Kevin Wang
Re: Math Challenge II-A 4.25 and 4.29 Help
by Dr. Kevin Wang - Saturday, 27 June 2020, 7:08 PM
 

For 4.25, the polynomial is $x^3 + 9x^2 + 26x + 24$.  To factor it, you can try Rational Root Theorem, and for any factor of the type $(x-a)$,  $a$ should be a factor of $24$.  Try a few factors of $24$, and as soon as you find one number that fits, you will be good shape.

For 4.29, The polynomial is $a^5 + a^4 + 1$.  your method is not bad.  Let's see where it leads:

$$1+a+a^2+a^3+a^4+a^5 = \frac{a^6-1}{a-1}=\frac{(a^3+1)(a^3-1)}{a-1}=(a^3+1)(a^2+a+1)$$

You may try to fill in the rest.