Question

Bill keeps a colony of honeybees on his property. He has created an expression to represent the number of honeybees that are in the colony t years after he decided to create the colony. The expression Bill created is -16t4 + 160t2 – 144.
After deciding to create the colony, it took a fair amount of time to prepare before actually getting bees in his colony. After some time, the population in the colony started to decrease; eventually Bill lost his colony.

How could you use Bill’s expression to find out when Bill first got his honeybees and when he lost his colony?




Factor out a negative 1 from the expression using a bracket. Then rewrite the first term inside the bracket as the square of a term. Do this so that if the term being squared were replaced with a variable, the term would have no coefficient.




Set the term being squared from part b equal to x and rewrite the expression in terms of x.




Factor the resulting expression inside the bracket.




Now that you have factored the expression, replace x with the term you initially substituted for.



When did Bill first get his honeybees and when did he lose them?



Use the same idea you used above to factor the expression 4x4 – 25y6.

Answer 1

Hello,

1) -16t^4+160t²-144=0
==>4t^4-40t²+36=0
==>4t^4-4t²-36t²+36=0
==>4(t^4-1)-36(t²-1)=0
==>4(t²-1)(t²+1)-36(t²-1)=0
==>4(t²-1)(t²-9)=0
==>(t+1)(t-1)(t+3)(t-3)=0
Bill first got his honeybees after 1 year and he lost his colony after 3 years.

2)
4x^4-25y^6=(2x²)²-(5y^3)²=(2x²-5y^3)(2x²+5y^3)