Could someone please check the first one and help me with the second one. I'm stuck-Thank you
1.What are theroots of the following polynomial equation?
(x+2i)(x-5)(x-2i)(x+8) = 0
I think the roots are -2i,5,2i,-8-you take out the roots as opposites of what are in the equaion correct?
2. Write a polynomial of the smallest degree with roots 3,3i and -3i
Final answer can't have parentheses
(x-3)(x-3i)(x+3i)=0 and then I'm stuck I don't know what to do when you multiply the x times the -3i
First find (x-3i)(x+3i)=x^2+9 and then
(x-3)(x^2+9)=x^3-3x^2+9x-27
Do the last two first: (x-3i)(x+3i) You can do FOIL (the anwer to x*(-3i) is -3xi), however, notice x+9 is the answer. (x^2+9) is the difference of two squares, x and 3i, so (x^2+9)=(x+3i)*(x-3i)
so you now have (x-3)(x^2+9)
= x^3+9x -3x^3-27 Using the FOIL method.
Whenn I get to (x-3)(x^2+9) and FOIL wouldn't it be x^3 +9x-3x^2 -27? I get (-3x^2) not (3x^3) when I multiply -3 times x^2 or am I missing something
Thank you
correc, typo. -3x^2
To find the roots of the polynomial equation (x+2i)(x-5)(x-2i)(x+8) = 0, you correctly spotted that the roots are the values that make the equation equal to zero. In this case, the roots are -2i, 5, 2i, and -8. You obtained the correct answer by taking out the roots as opposites of what is present in the equation.
Now, let's move on to the second question. You correctly started with the roots 3, 3i, and -3i and formed the equation (x-3)(x-3i)(x+3i) = 0. To proceed further, you need to multiply the terms together.
To do that, you can use the distributive property. Start by multiplying the first two terms, (x-3) and (x-3i), then multiply the result by (x+3i).
(x-3)(x-3i) = x(x) + x(-3i) - 3(x) - 3(-3i) = x^2 - 3ix - 3x + 9i = x^2 - (3x + 3i)x + 9i
Now, multiply the obtained result by the third term, (x+3i).
(x^2 - (3x + 3i)x + 9i)(x+3i) = x^2(x+3i) - (3x + 3i)x(x+3i) + 9i(x+3i)
Using the distributive property again, expand each expression:
x^2(x+3i) = x^3 + 3ix^2
-(3x + 3i)x(x+3i) = -(3x + 3i) * (x^2 + 3ix)
9i(x+3i) = 9ix + 27i^2
Simplifying further, combining like terms:
x^3 + 3ix^2 - 3x^3 - 9ix^2 - 9x - 9ix + 9ix + 27i^2
Next, simplify the terms involving i, keeping in mind that i^2 is equal to -1:
x^3 + 3ix^2 - 3x^3 - 9ix^2 - 9x + 9ix + 9ix - 27
Rearranging the terms:
(x^3 - 3x^3) + (3ix^2 - 9ix^2) + (-9x + 9ix + 9ix) - 27
Combining like terms:
-2x^3 - 6ix^2 + 18ix - 27
So, the polynomial of the smallest degree with roots 3, 3i, and -3i is:
-2x^3 - 6ix^2 + 18ix - 27