Question

solve\
t⁴ - 3t³ - 40t² = 0\
find all solutions other than 0 and fill in the boxes:\
t = 0, \boxed{ }, \boxed{ }\
hint: factor the gcf then factor the trinomial

Explanation:

Step1: Factor out GCF

The greatest common factor (GCF) of \(t^4\), \(-3t^3\), and \(-40t^2\) is \(t^2\). So we factor out \(t^2\):
\(t^2(t^2 - 3t - 40)=0\)

Step2: Factor the trinomial

We need to factor the quadratic trinomial \(t^2 - 3t - 40\). We look for two numbers that multiply to \(-40\) and add up to \(-3\). The numbers are \(-8\) and \(5\) because \((-8)\times5 = -40\) and \(-8 + 5=-3\). So we can factor the trinomial as:
\(t^2 - 3t - 40=(t - 8)(t + 5)\)

Step3: Find the solutions

From the factored form \(t^2(t - 8)(t + 5)=0\), we set each factor equal to zero. We already know \(t = 0\) is a solution from \(t^2 = 0\). For the other factors:

• For \(t - 8 = 0\), we get \(t = 8\).
• For \(t + 5 = 0\), we get \(t=-5\).

Answer:

\(t = 0\), \(t = 8\), \(t=-5\)