Using Other Complex Numbers

Oct 13, 2025 by magicalceylontours.com 30 views

Understanding the Function $f(x)$

The given function is $f(x) = x^3 - 2x^2$ . This is a polynomial function of degree 3, which means it has a cubic term and a quadratic term. To find the equivalent expression for $f(t)$ , we need to substitute $t$ into the function in place of $x$ .

Substituting $t$ into the Function

When we substitute $t$ into the function, we get:

$f(t) = t^3 - 2t^2$

Evaluating the Expression

To evaluate the expression, we need to find the value of $t$ that makes the expression equivalent to one of the given options. However, the value of $t$ is not provided, and we are only given the options to choose from.

Analyzing the Options

Let's analyze each option and see if we can find a value of $t$ that makes the expression equivalent to it.

Option A: $-2+i$

To find the value of $t$ that makes the expression equivalent to $-2+i$ , we need to set up an equation:

$t^3 - 2t^2 = -2+i$

We can start by factoring out $t^2$ from the left-hand side:

$t^2(t - 2) = -2+i$

Now, we can see that $t^2$ must be equal to $-2+i$ divided by $(t-2)$ . However, this is not possible since $t^2$ is always non-negative, and $-2+i$ divided by $(t-2)$ is not always non-negative.

Option B: $-2-i$

Similarly, to find the value of $t$ that makes the expression equivalent to $-2-i$ , we need to set up an equation:

$t^3 - 2t^2 = -2-i$

We can start by factoring out $t^2$ from the left-hand side:

$t^2(t - 2) = -2-i$

Now, we can see that $t^2$ must be equal to $-2-i$ divided by $(t-2)$ . However, this is not possible since $t^2$ is always non-negative, and $-2-i$ divided by $(t-2)$ is not always non-negative.

Option C: $2+i$

To find the value of $t$ that makes the expression equivalent to $2+i$ , we need to set up an equation:

$t^3 - 2t^2 = 2+i$

We can start by factoring out $t^2$ from the left-hand side:

$t^2(t - 2) = 2+i$

Now, we can see that $t^2$ must be equal to $2+i$ divided by $(t-2)$ . However, this is not possible since $t^2$ is always non-negative, and $2+i$ divided by $(t-2)$ is not always non-negative.

Option D: $2-i$

Similarly, to find the value of $t$ that makes the expression equivalent to $2-i$ , we need to set up an equation:

$t^3 - 2t^2 = 2-i$

We can start by factoring out $t^2$ from the left-hand side:

$t^2(t - 2) = 2-i$

Now, we can see that $t^2$ must be equal to $2-i$ divided by $(t-2)$ . However, this is not possible since $t^2$ is always non-negative, and $2-i$ divided by $(t-2)$ is not always non-negative.

Conclusion

Based on the analysis above, we can conclude that none of the options are equivalent to $f(t)$ . However, we can try to find a value of $t$ that makes the expression equivalent to one of the options.

Finding a Value of $t$

Let's try to find a value of $t$ that makes the expression equivalent to one of the options. We can start by setting $t = 2$ :

$f(2) = 2^3 - 2(2)^2 = 8 - 8 = 0$

Now, we can see that $f(2) = 0$ , which is not equal to any of the options.

Using Complex Numbers

Since we are not getting any real solutions, let's try to use complex numbers. We can start by setting $t = 2 + i$ :

$f(2+i) = (2+i)^3 - 2(2+i)^2$

We can expand the expression using the binomial theorem:

$f(2+i) = (2+i)^3 - 2(2+i)^2$

$f(2+i) = (2+i)(2+i)(2+i) - 2(2+i)(2+i)$

$f(2+i) = (2+i)(4+4i+i^2) - 2(4+4i+i^2)$

$f(2+i) = (2+i)(3+4i) - 2(3+4i)$

$f(2+i) = 6+8i+3i+4i^2 - 6-8i$

$f(2+i) = 6+11i-4 - 6-8i$

$f(2+i) = 2+3i$

Now, we can see that $f(2+i) = 2+3i$ , which is not equal to any of the options.

Using Other Complex Numbers

Let's try to use other complex numbers. We can start by setting $t = 2-i$ :

$f(2-i) = (2-i)^3 - 2(2-i)^2$

We can expand the expression using the binomial theorem:

$f(2-i) = (2-i)^3 - 2(2-i)^2$

$f(2-i) = (2-i)(2-i)(2-i) - 2(2-i)(2-i)$

$f(2-i) = (2-i)(4-4i+i^2) - 2(4-4i+i^2)$

$f(2-i) = (2-i)(3-4i) - 2(3-4i)$

$f(2-i) = 6-8i-3i+4i^2 - 6+8i$

$f(2-i) = 6-11i-4 - 6+8i$

$f(2-i) = 2-3i$

Now, we can see that $f(2-i) = 2-3i$ , which is equal to option D.

Conclusion

Based on the analysis above, we can conclude that the expression equivalent to $f(t)$ is option D: $2-i$ .

Final Answer

The final answer is option D: $2-i$ .

Understanding the Function $f(x)$

Q: What is the equivalent expression for $f(t)$ ?

A: The equivalent expression for $f(t)$ is $t^3 - 2t^2$ .

Q: How do we find the value of $t$ that makes the expression equivalent to one of the options?

A: We can start by setting up an equation using the given options. For example, if we want to find the value of $t$ that makes the expression equivalent to option A: $-2+i$ , we can set up the equation $t^3 - 2t^2 = -2+i$ .

Q: Why can't we find a real value of $t$ that makes the expression equivalent to one of the options?

A: We can't find a real value of $t$ that makes the expression equivalent to one of the options because the expression is a cubic function, and the cubic term $t^3$ will always be non-zero. This means that the expression will never be equal to a real number.

Q: Can we use complex numbers to find the equivalent expression for $f(t)$ ?

A: Yes, we can use complex numbers to find the equivalent expression for $f(t)$ . By setting $t = 2 + i$ , we can find the equivalent expression for $f(t)$ .

Q: What is the equivalent expression for $f(t)$ when $t = 2 + i$ ?

A: The equivalent expression for $f(t)$ when $t = 2 + i$ is $2+3i$ .

Q: Why is this not equal to any of the options?

A: This is not equal to any of the options because the expression $2+3i$ is a complex number, and the options are all real numbers.

Q: Can we use other complex numbers to find the equivalent expression for $f(t)$ ?

A: Yes, we can use other complex numbers to find the equivalent expression for $f(t)$ . By setting $t = 2 - i$ , we can find the equivalent expression for $f(t)$ .

Q: What is the equivalent expression for $f(t)$ when $t = 2 - i$ ?

A: The equivalent expression for $f(t)$ when $t = 2 - i$ is $2-3i$ .

Q: Is this equal to any of the options?

A: Yes, this is equal to option D: $2-i$ .

Q: What is the final answer?

A: The final answer is option D: $2-i$ .

Q: Why is this the final answer?

A: This is the final answer because we have shown that the expression equivalent to $f(t)$ is $2-3i$ , and this is equal to option D: $2-i$ .

Conclusion

In conclusion, we have shown that the expression equivalent to $f(t)$ is option D: $2-i$ . This is the final answer.

Final Answer

The final answer is option D: $2-i$ .