Write a quartic function that has roots that are 3+5i, -4 and 7 and

f(-3)=53.

I have tried it a bunch of times and can't seem to get it.

(x-3+5i)(x-3-5i)(x+4)(x-7)
I multiplied them together
x^4-9x^3+24x^2+66x-952
I was a little unsure what f(-3)=53 means.

I thought I should plug -3 in for x and it should equal 53. That doesn't work.

Can you please help me see where I went wrong?

Ahh, how would you know that the function wasn't

f(x) = a(x-3+5i)(x-3-5i)(x+4)(x-7)
= a(x^4-9x^3+24x^2+66x-952)
Wouldn't it still have those same roots if you set f(x) = 0 ?

I am going to trust that you expanded correctly.
Now let f(-3) = 53
53 = a((-3)^4 - 9(-3)^3 + 24(-3)^2 + 66(-3) - 952)

solve for a, then multiply your expanded expression by that value.
Take over...

How do you know that it is supposed to have an a in front?

I get where a comes from. Is the answer 53/-610? It doesn't seem right.

I got -53/610 also

" How do you know that it is supposed to have an a in front? "

Bob, I guess the answer was blowing in the wind.

To find a quartic function that satisfies the given conditions, we can begin by using the roots to write out the factors of the function.

The roots are given as 3+5i, 3-5i, -4, and 7. Remember that complex roots always come in conjugate pairs. So, we have the factors:

(x - (3 + 5i))(x - (3 - 5i))(x - (-4))(x - 7)

To simplify this expression, we can start by multiplying the complex conjugate factors separately:

(x - 3 - 5i)(x - 3 + 5i) = (x - 3)^2 - (5i)^2 = x^2 - 6x + 9 + 25 = x^2 - 6x + 34

Now we can rewrite the quartic function with the simplified factors:

(x^2 - 6x + 34)(x + 4)(x - 7)

To avoid confusion, we can rewrite this as:

(x^2 - 6x + 34)(x + 4)(x - 7) * 1

Finally, we can multiply all the factors together to get the expanded form of the quartic function:

(x^2 - 6x + 34)(x + 4)(x - 7) = x^4 - 6x^3 + 34x^2 + 4x^3 - 24x^2 + 136x - 7x^2 + 42x - 238

Combining like terms, we have:

x^4 - 2x^3 + 3x^2 + 178x - 238

Now let's address the second part of the problem, finding the value of f(-3) = 53.

To determine if the function satisfies this condition, we can substitute -3 into the function and check if it equals 53:

f(-3) = (-3)^4 - 2(-3)^3 + 3(-3)^2 + 178(-3) - 238
= 81 + 54 + 27 - 534 - 238
= -600

So, the function we found does not satisfy f(-3) = 53.

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