Algebra
Calculus
Trigonometry
Matrix
Question
Factor the expression
e2t(et−1)(e4t+e3t+e2t+et+1)
Evaluate
e7t−e2t
Factor the expression
e2t(e5t−1)
Solution
More Steps
Calculate
e5t−1
Evaluate
(et)5−1
Calculate
(et)5+(et)4+(et)3+(et)2+et−(et)4−(et)3−(et)2−et−1
Rewrite the expression
et(et)4+et(et)3+et(et)2+et×et+et−(et)4−(et)3−(et)2−et−1
Factor out et from the expression
et((et)4+(et)3+(et)2+et+1)−(et)4−(et)3−(et)2−et−1
Factor out −1 from the expression
et((et)4+(et)3+(et)2+et+1)−((et)4+(et)3+(et)2+et+1)
Factor out (et)4+(et)3+(et)2+et+1 from the expression
(et−1)((et)4+(et)3+(et)2+et+1)
Factor the expression
(et−1)(e4t+e3t+e2t+et+1)
e2t(et−1)(e4t+e3t+e2t+et+1)
Show Solution
Find the roots
t=0
Evaluate
e7t−e2t
To find the roots of the expression,set the expression equal to 0
e7t−e2t=0
Factor the expression
More Steps
Factor the expression
e7t−e2t
Evaluate
(et)7−(et)2
Rewrite the expression
(et)2(et)5−(et)2
Factor out (et)2 from the expression
(et)2((et)5−1)
Factor the expression
More Steps
Evaluate
(et)5−1
Calculate
(et)5+(et)4+(et)3+(et)2+et−(et)4−(et)3−(et)2−et−1
Rewrite the expression
et(et)4+et(et)3+et(et)2+et×et+et−(et)4−(et)3−(et)2−et−1
Factor out et from the expression
et((et)4+(et)3+(et)2+et+1)−(et)4−(et)3−(et)2−et−1
Factor out −1 from the expression
et((et)4+(et)3+(et)2+et+1)−((et)4+(et)3+(et)2+et+1)
Factor out (et)4+(et)3+(et)2+et+1 from the expression
(et−1)((et)4+(et)3+(et)2+et+1)
(et)2(et−1)((et)4+(et)3+(et)2+et+1)
Factor the expression
(et)2(et−1)(e4t+e3t+e2t+et+1)
(et)2(et−1)(e4t+e3t+e2t+et+1)=0
Separate the equation into 3 possible cases
(et)2=0et−1=0e4t+e3t+e2t+et+1=0
Since the left-hand side is always positive,and the right-hand side is always 0,the statement is false for any value of t
t∈/Ret−1=0e4t+e3t+e2t+et+1=0
Solve the equation
More Steps
Evaluate
et−1=0
Move the constant to the right-hand side and change its sign
et=0+1
Removing 0 doesn't change the value,so remove it from the expression
et=1
Write the number in exponential form with the base of e
et=e0
Since the bases are the same,set the exponents equal
t=0
t∈/Rt=0e4t+e3t+e2t+et+1=0
Since the left-hand side is always positive,and the right-hand side is always 0,the statement is false for any value of t
t∈/Rt=0t∈/R
Solution
t=0
Show Solution