Creating polynomials

Question

Create a polynomial that fits the following criteria:

1. A degree of 4
2. A solution at 2 + 4i
3. Two solutions at x = -3

Answer 1

x= 2+4i x=2-4i x=-3 x =-3

imaginary roots come in conjugate pairs, if 2+4i is a root, then so is 2-4i

corresponding factors are (x-2-4i)(x-2+4i)(x+3)(x+3)

multiply them together and you get the 4th degree polynomial

= (x^2+6x+9)(x^2 -4x+ 20)

= x^4 + 2x^3 + 5x^2 +84x + 180

Answer 2

Step 1: Lets Understand the question first

1. Degree of 4 polynomial has 4 roots.
2. One root is 2+4i. Its conjugate,2−4i, is also a root because complex roots come in pairs.
3. Another root is x=−3, but we have to get two solutions. That means it has multiplicity 2.

Step 2: Factors

Each root corresponds to a factor:

1. For 2+4i : (x−(2+4i))
2. For 2−4i : (x−(2−4i))
3. For x=−3 (repeated): (x + 3)²

So the polynomial is:

P(x)=(x−(2+4i))(x−(2−4i))(x+3)²

now lets simplify the above equation separatel

{ (x−(2+4i))(x−(2−4i)) }

so for now we are excluding and not multiplying (x+3)²

(x−(2+4i))(x−(2−4i)) = ((x−2)+4i)((x−2)+4i)

[ Applying(a−b)(a+b)=a²−b² Here a= (x−2) b=4i ]

= (x−2)²−(4i)²

Expand (x−2)² = x²−4x+4 based on formula (a-b)²=a²-2ab+b²

(4i)²=−16

= (x²−4x+4 )-(-16)

= (x²−4x+4 +16)

= x²−4x+20

So:

(x−(2+4i))(x−(2−4i))=x²−4x+20

Now the polynomial is:

P(x)=(x2−4x+20)(x+3)²

Expand (x+3)²=x2+6x+9 based on formula (a+b)²=a²⁺2ab+b²

P(x)=(x2−4x+20)(x2+6x+9)

P(x)= x²(x²+6x+9)−4x(x²+6x+9)+20(x²+6x+9)

= x⁴+6x³+9x²−4x³−24x²−36x+20x²+120x+180

Add now,

P(x)=x⁴+2x³+5x²+84x+180

Final Answer:

P(x)=x⁴+2x³+5x²+84x+180

Answer 3

When creating a polynomial that you already know the solutions, you just have to write it in factored form with each binomial representing one of the solutions. Since -3 will be a solution twice, (x-(-3)) and (x-(-3) will be two of the binomials. The other solution that must be included is 2+4i, and since it has to be a 4th degree polynomial, the 4th solution will be the conjugate of that, so 2-4i.

So If we put it all together it looks like this:

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

That may be a sufficient answer in that form, but if you need it in standard form, you can just multiply and simplify.

If you need it in that form it would be:

f(x) = x⁴ + 2x³ + 5x² + 84x + 180

Answer 4

P(x) = (x - (2 + 4i))(x - (2 - 4i))(x - (-3))²