Which polynomial function has a leading coefficient of 1 and roots 2i and 3i with multiplicity 1?
f(x) = (x – 2i)(x – 3i)
f(x) = (x + 2i)(x + 3i)
f(x) = (x – 2)(x – 3)(x – 2i)(x – 3i)
f(x) = (x + 2i)(x + 3i)(x – 2i)(x – 3i)

Step-by-step explanation:

Let's check each case to determine the solution to the problem

case A

The roots of this polynomial are

------> with multiplicity

------> with multiplicity

leading coefficient of

therefore

the case A is a solution

case B

The roots of this polynomial are

------> with multiplicity

------> with multiplicity

leading coefficient of

therefore

the case B is not a solution

case C

The roots of this polynomial are

------> with multiplicity

------> with multiplicity

------> with multiplicity

------> with multiplicity

leading coefficient of

therefore

the case C is a solution

case D

The roots of this polynomial are

------> with multiplicity

------> with multiplicity

------> with multiplicity

------> with multiplicity

leading coefficient of

therefore

the case D is  a solution

The correct option is (D)

Step-by-step explanation:  We are given to select the correct polynomial function that has a leading co-efficient of 1 and roots 2i and 3i with multiplicity 1.

We know that the complex roots of a polynomial function always occur in pairs.

That is, if (a + bi) is a root of a polynomial function P(x), then its conjugate (a - bi) will also be a root of P(x).

Since, 2i and 3i are the roots of the polynomial function, so their conjugates - 2i and - 3i will also be the roots.

Also, we know that if x = a is aroot of a polynomial, then (x - a) is a factor of the polynomial.

Therefore, (x -2i), (x - 3i), (x + 2i) and (x + 3i) are the factors of the given polynomial.

Also, since the leading co-efficient is 1, so the required polynomial will be

Thus, (D) is the correct option.

answer: ratio of volume of cone (with 3 times the height and equal diameter) to volume of cylinder is 2 to 1. the value of π is not used.

step-by-step explanation:

radius is diameter/2 = 4.

area of circle is π r^2.

volume of cylinder is height × area of base = hπr^2 = 3(4^2) π = 48π cubic inches.

not knowing the formula for volume of cone, i calculate it using calculus (which is for calculating things).

volume of cone is integral 0 to h area(y) dy. area(y) is π r(y)^2. r(y) runs r to zero as y runs zero to h, r(0) = r, r(h) = 0.

r(y) = r - r/h y

area(y) = π (r - r/h y)^2 = π (r^2 - 2r^2y/h + r^2y^2/h^2) = π r^2(1-2y/h+y^2/h^2)

antiderivative area(y) = π r^2 (y-y^2/h+y^3/(3h^2))

integral 0 to h = π r^2(h-h^2/h+h^3/(3h^2) - (0)) = π r^2(h - h + h/3) = π r^2(h/3) = π/3 r^2 h

volume of this cone is π/3(4^2)18 cubic inches.

= 6 16 π = 96π cubic inches.

cone has twice the volume, 96/48.

cone height is 6 times the cylinder.

ratio of volume formula: π r^2 cancels.

cone over cylinder ratio is 6h/3 to h, 2 to 1.

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step-by-step explanation: