To find the polynomial with rational coefficients whose roots are \(0, 0, 2, 2, -2, -2\), we can follow these steps:
### Step 1: Identify the roots
The roots given are:
- \(0\) (with multiplicity 2)
- \(2\) (with multiplicity 2)
- \(-2\) (with multiplicity 2)
### Step 2: Write the factors corresponding to the roots
For each root, we can write a factor of the polynomial:
- For the root \(0\), the factor is \(x\) (since it has multiplicity 2, we write \(x^2\)).
- For the root \(2\), the factor is \((x - 2)\) (with multiplicity 2, we write \((x - 2)^2\)).
- For the root \(-2\), the factor is \((x + 2)\) (with multiplicity 2, we write \((x + 2)^2\)).
Thus, the polynomial can be expressed as:
\[
P(x) = x^2 (x - 2)^2 (x + 2)^2
\]
### Step 3: Expand the factors
First, we expand \((x - 2)^2\) and \((x + 2)^2\):
\[
(x - 2)^2 = x^2 - 4x + 4
\]
\[
(x + 2)^2 = x^2 + 4x + 4
\]
Next, we multiply these two results:
\[
(x - 2)^2 (x + 2)^2 = (x^2 - 4x + 4)(x^2 + 4x + 4)
\]
### Step 4: Use the distributive property to expand
Now, we will expand \((x^2 - 4x + 4)(x^2 + 4x + 4)\):
\[
= x^2(x^2 + 4x + 4) - 4x(x^2 + 4x + 4) + 4(x^2 + 4x + 4)
\]
Calculating each term:
1. \(x^2(x^2 + 4x + 4) = x^4 + 4x^3 + 4x^2\)
2. \(-4x(x^2 + 4x + 4) = -4x^3 - 16x^2 - 16x\)
3. \(4(x^2 + 4x + 4) = 4x^2 + 16x + 16\)
Now, combine all these:
\[
x^4 + 4x^3 + 4x^2 - 4x^3 - 16x^2 - 16x + 4x^2 + 16x + 16
\]
### Step 5: Combine like terms
Combining the like terms:
\[
x^4 + (4x^3 - 4x^3) + (4x^2 - 16x^2 + 4x^2) + (-16x + 16x) + 16
\]
This simplifies to:
\[
x^4 + 0x^3 - 8x^2 + 0x + 16 = x^4 - 8x^2 + 16
\]
### Step 6: Include the factor \(x^2\)
Now, we multiply this result by \(x^2\) (from the root \(0\)):
\[
P(x) = x^2(x^4 - 8x^2 + 16) = x^6 - 8x^4 + 16x^2
\]
### Final polynomial
Thus, the polynomial with rational coefficients whose roots are \(0, 0, 2, 2, -2, -2\) is:
\[
P(x) = x^6 - 8x^4 + 16x^2
\]
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