To solve the equation:
[tex]\[ x^4 - 4x^3 = 6x^2 - 12x \][/tex]
we start by setting it to zero:
[tex]\[ x^4 - 4x^3 - 6x^2 + 12x = 0 \][/tex]
Now, let's find the roots.
We can factor out the greatest common factor (GCF) first. Looking at the terms, we notice that each term has an [tex]\( x \)[/tex] as a factor:
[tex]\[ x(x^3 - 4x^2 - 6x + 12) = 0 \][/tex]
From this factorization, we immediately see one root:
[tex]\[ x = 0 \][/tex]
To find the remaining roots, we focus on the cubic polynomial:
[tex]\[ x^3 - 4x^2 - 6x + 12 = 0 \][/tex]
To solve this cubic equation, a systematic way would be to either use synthetic division to factor out the polynomial further or use a graphing calculator to approximate the roots.
Using a graphing calculator for these steps:
1. Plot the graph of [tex]\( y = x^3 - 4x^2 - 6x + 12 \)[/tex].
2. Identify the points where the graph intersects the x-axis.
After graphing, we can find that the polynomial [tex]\(x^3 - 4x^2 - 6x + 12\)[/tex] has the following integer roots:
[tex]\[ x = -2 \][/tex]
Combining all roots, we have the complete set: [tex]\( x = 0 \)[/tex] and [tex]\( x = -2 \)[/tex].
Thus, the integral roots of the equation [tex]\( x^4 - 4x^3 - 6x^2 + 12x = 0 \)[/tex] from least to greatest are:
[tex]\[
-2 \quad \text{and} \quad 0
\][/tex]
