# Thread: [SOLVED] write an equation of a polynomial

1. ## [SOLVED] write an equation of a polynomial

Write an equation of a polynomial of degree 4 and integer coefficients with roots -2i, 1/3 and 5.

2. Hello,
Originally Posted by moonman
Write an equation of a polynomial of degree 4 and integer coefficients with roots -2i, 1/3 and 5.
If a is a root of a polynomial, then x-a divides this polynomial.
So you can say that (x-(-2i))(x-1/3)(x-5) divides the polynomial.

What about the 4th factor ? Since the coefficients are integers, if a complex number (like -2i) is a root, then its conjugate is also a root.

The conjugate of a complex number a+ib is a-ib. Here, a=0 and b=-2. Can you do it ?

3. Originally Posted by moonman
Write an equation of a polynomial of degree 4 and integer coefficients with roots -2i, 1/3 and 5.
1. I assume that this polynomial has +2i as a root too.

2. $p(x)=a\left((x^2+4)(x-5)\left(x-\frac13\right)\right)$

3. Expand the brackets:

$p(x)=a\left(x^4-\frac{16}3 x^3 + \frac{17}3 x^2 - \frac{64}3 x + \frac{20}3 \right)$

4. Choose a = 3 and you'll get:

$p(x)=3x^4-16x^3+17x^2-64x+20$

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EDIT: Too fast for me, Moo

4. Originally Posted by earboth
1. I assume that this polynomial has +2i as a root too.

2. $p(x)=a\left((x^2+4)(x-5)\left(x-\frac13\right)\right)$

3. Expand the brackets:

$p(x)=a\left(x^4-\frac{16}3 x^3 + \frac{17}3 x^2 - \frac{64}3 x + \frac{20}3 \right)$

4. Choose a = 3 and you'll get:

$p(x)=3x^4-16x^3+17x^2-64x+20$

-----------------------------------------------------------

EDIT: Too fast for me, Moo

In step 2; can you show me how you got (x^2 + 4)

5. Originally Posted by moonman
In step 2; can you show me how you got (x^2 + 4)
If the roots of the polynomial are $x = -2i~\vee~x=+2i$ then you have the product:

$(x-2i)(x+2i)=x^2-4i^2=x^2-4\cdot (-1)=x^2+4$

6. Originally Posted by earboth
If the roots of the polynomial are $x = -2i~\vee~x=+2i$ then you have the product:

$(x-2i)(x+2i)=x^2-4i^2=x^2-4\cdot (-1)=x^2+4$

awesome