To find the roots of the polynomial equation [tex]\(x^4 + x^3 = 4x^2 + 4x\)[/tex], we will systematically solve it step by step.
1. Rewrite the equation:
Start with the given polynomial equation:
[tex]\[
x^4 + x^3 = 4x^2 + 4x
\][/tex]
2. Bring all terms to one side:
To facilitate solving, we move all terms to one side of the equation:
[tex]\[
x^4 + x^3 - 4x^2 - 4x = 0
\][/tex]
3. Factor the polynomial:
To solve for [tex]\( x \)[/tex], we can try to factor the polynomial. Notice that all terms have a common factor, which is [tex]\( x \)[/tex]:
[tex]\[
x(x^3 + x^2 - 4x - 4) = 0
\][/tex]
This gives us one root directly:
[tex]\[
x = 0
\][/tex]
4. Solve the remaining polynomial:
Now, we need to solve the cubic equation:
[tex]\[
x^3 + x^2 - 4x - 4 = 0
\][/tex]
Through further factoring or using methods such as the Rational Root Theorem, we can find that the factors of this polynomial include [tex]\( (x + 2) \)[/tex], [tex]\( (x + 1) \)[/tex], and [tex]\( (x - 2) \)[/tex].
This means we can write:
[tex]\[
x^3 + x^2 - 4x - 4 = (x + 2)(x + 1)(x - 2)
\][/tex]
5. Find the remaining roots:
Solve for [tex]\( x \)[/tex] from the factored form:
[tex]\[
(x + 2) = 0 \quad \Rightarrow \quad x = -2
\][/tex]
[tex]\[
(x + 1) = 0 \quad \Rightarrow \quad x = -1
\][/tex]
[tex]\[
(x - 2) = 0 \quad \Rightarrow \quad x = 2
\][/tex]
6. Collect all roots:
Combining these solutions, we get the roots of the polynomial equation [tex]\( x^4 + x^3 = 4x^2 + 4x \)[/tex]:
[tex]\[
x = -2, -1, 0, 2
\][/tex]
Therefore, the roots of the given equation [tex]\( x^4 + x^3 = 4x^2 + 4x \)[/tex] are:
[tex]\[
\boxed{-2, -1, 0, 2}
\][/tex]
