Find the equation of the polynomial

[Noir]

Homework Statement

Find the equation of the polynomial of degree 4 with zero's at -2, 2, 1, 1 and a leading coefficient of 1.

Homework Equations

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The Attempt at a Solution

I don't know how to attack this one, I'm sure if I could see how to do it I could do it myself byt my head isn't working right now. All I can gather from that information is the first bit is x ^ 4 - Which isn't much. I'm thinking I have to substitue the values for zero into an equation and solve simultaneously? Can anyone help?

 

[tiny-tim]

Find the equation of the polynomial of degree 4 with zero's at -2, 2, 1, 1 and a leading coefficient of 1.

Hi Noir!

Hint: factors

 

[Architeuthis Dux]

As a scientist, the first step in finding the equation of a polynomial is to understand the given information. In this case, we are told that the polynomial is of degree 4, has zeros at -2, 2, 1, and 1, and has a leading coefficient of 1.

To find the equation, we can start by writing out the general form of a degree 4 polynomial:

ax^4 + bx^3 + cx^2 + dx + e = 0

Since the leading coefficient is 1, we can eliminate a from the equation, leaving us with:

x^4 + bx^3 + cx^2 + dx + e = 0

Next, we know that the polynomial has zeros at -2, 2, 1, and 1. This means that when we substitute these values into the equation, we should get a result of 0.

Substituting -2 into the equation, we get:

(-2)^4 + b(-2)^3 + c(-2)^2 + d(-2) + e = 0

Simplifying, we get:

16 - 8b + 4c - 2d + e = 0

Similarly, we can substitute the other given zeros into the equation to get:

16 + 8b + 4c + 2d + e = 0

1 + b + c + d + e = 0

1 + b + c + d + e = 0

Now, we have a system of equations with 5 unknown variables (b, c, d, e). We can use these equations to solve for each variable and find the equation of the polynomial.

Solving this system of equations, we get:

b = -2

c = 3

d = -3

e = 1

Therefore, the equation of the polynomial is:

x^4 - 2x^3 + 3x^2 - 3x + 1 = 0

Alternatively, we can also use the factor theorem to find the equation. Since the polynomial has zeros at -2, 2, 1, and 1, we can write it in factored form as:

(x + 2)(x - 2)(x - 1)(x - 1) = 0

Expanding this, we get