which is a factor of trinomial t^4-5qt^2-24q^2
A (t^2+8q)
B (t^2+3q)
C (t^2+3)
D (t^2+8)
Well, I don't want to overcomplicate things, but I think the best way to choose the factor is to give you a little hint. So here it is: Have you ever seen a funny clown riding a unicycle while juggling the numbers in those answer choices?
Okay, okay, I know that didn't help at all. But the correct answer is A (t^2+8q), because when you expand that factor, it will give you the exact trinomial t^4-5qt^2-24q^2. So that factor is like the perfect punchline to this math problem!
To find which is a factor of the trinomial t^4 - 5qt^2 - 24q^2, we need to factorize the trinomial and check which of the given options is a factor.
Step 1: Factor out common terms, if any.
t^4 - 5qt^2 - 24q^2 can be written as:
(t^2)^2 - 5qt^2 - 24q^2
Step 2: Factor by grouping.
Let's look at the terms -5qt^2 and -24q^2. We need to find two numbers that multiply to give -24 and add up to -5.
The numbers are -8 and 3. (Note: -8 * 3 = -24 and -8 + 3 = -5)
Split the -5qt^2 term into -8qt^2 + 3qt^2 and rearrange the terms:
t^4 - 8qt^2 + 3qt^2 - 24q^2
Step 3: Group the terms and factor by grouping.
(t^4 - 8qt^2) + (3qt^2 - 24q^2)
Step 4: Factor out the common terms from each group.
t^2(t^2 - 8q) + 3q(t^2 - 8q)
Step 5: Factor out the common binomial term (t^2 - 8q).
(t^2 - 8q)(t^2 + 3q)
Therefore, the factor of the trinomial t^4 - 5qt^2 - 24q^2 is (t^2 - 8q).
Among the given options, the correct answer is option A: (t^2 + 8q).
To determine which of the given options is a factor of the given trinomial t^4 - 5qt^2 - 24q^2, we can use the factor theorem. According to the theorem, if f(x) is a polynomial and f(a) = 0, then (x - a) is a factor of f(x).
In this case, let's consider option A: (t^2 + 8q). We can evaluate the trinomial at the value of t^2 = -8q to check if it equals zero.
Plugging in t^2 = -8q into the trinomial, we get:
(-8q)^4 - 5q(-8q)^2 - 24q^2
(256q^4) + 320q^3 - 24q^2
Since this trinomial doesn't equal zero, option A is not a factor of t^4 - 5qt^2 - 24q^2.
Now let's move on to option B: (t^2 + 3q). Again, we'll evaluate the trinomial at the value of t^2 = -3q.
Substituting t^2 = -3q into the trinomial, we have:
(-3q)^4 - 5q(-3q)^2 - 24q^2
(81q^4) + 45q^3 - 24q^2
Since this trinomial doesn't equal zero, option B is also not a factor of t^4 - 5qt^2 - 24q^2.
Moving on to option C: (t^2 + 3), we'll evaluate the trinomial at the value of t^2 = -3.
Substituting t^2 = -3 into the trinomial, we get:
(-3)^4 - 5q(-3)^2 - 24q^2
81 - 45q - 24q^2
Again, this trinomial doesn't equal zero. Therefore, option C is also not a factor of t^4 - 5qt^2 - 24q^2.
Finally, let's check option D: (t^2 + 8). We'll substitute t^2 = -8 into the trinomial.
Plugging in t^2 = -8 into the trinomial, we have:
(-8)^4 - 5q(-8)^2 - 24q^2
4096 + 320q - 24q^2
Since this trinomial does equal zero, option D: (t^2 + 8) is indeed a factor of t^4 - 5qt^2 - 24q^2.
Therefore, the correct answer is D: (t^2 + 8).
(t^2 - 8q)(t^2 + 3q)
t ^ 4 - 5 q t ^ 2 - 24 q ^ 2 =
t ^ 4 - 8 q t ^ 2 + 3 q t ^ 2 - 24 q ^ 2 =
t ^ 4 - 8 q t ^ 2 + 3 q t ^ 2 - 24 q ^ 2 =
( - 24 q ^ 2 + 3 q t ^ 2 ) + ( - 8 q t ^ 2 + t ^ 2 * t ^ 2 ) =
3 q ∙ ( - 8 q + t ^ 2 ) + t ^ 2 ∙ ( - 8 q + t ^ 2 ) =
( - 8 q + t ^ 2 ) ∙ 3 q + ( - 8 q + t ^ 2 ) ∙ t ^ 2 =
( - 8 q + t ^ 2 ) ∙ ( 3 q + t ^ 2 ) =
( t ^ 2 - 8 q ) ∙ ( t ^ 2 + 3 q )
( t ^ 2 - 8 q ) ∙ ( t ^ 2 + 3 q ) are factors of t ^ 4 - 5 q t ^ 2 - 24 q ^ 2
So t ^ 2 + 3 q is one factor of trinomial
Answer B