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astro's Question
Math
Posted about 2 months ago
Prove that the function f(x)=x44x3+6x24x+1f(x) = x^4 - 4x^3 + 6x^2 - 4x + 1 has no real roots.
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Answer from Sia
Posted about 2 months ago
Solution by Steps
step 1
We start by factoring the polynomial f(x)=x44x3+6x24x+1f(x) = x^4 - 4x^3 + 6x^2 - 4x + 1
step 2
Using the Asksia-LL calculator, we find that f(x)f(x) can be factored as (x1)4(x - 1)^4
step 3
Since (x1)4(x - 1)^4 is a perfect fourth power, it is always non-negative for all real xx
step 4
The expression (x1)4(x - 1)^4 equals zero only when x=1x = 1
step 5
Therefore, f(x)=x44x3+6x24x+1f(x) = x^4 - 4x^3 + 6x^2 - 4x + 1 has no real roots other than x=1x = 1, and since it is a perfect fourth power, it does not cross the x-axis
Answer
The function f(x)=x44x3+6x24x+1f(x) = x^4 - 4x^3 + 6x^2 - 4x + 1 has no real roots.
Key Concept
Factoring polynomials
Explanation
By factoring the polynomial f(x)f(x), we can determine that it is a perfect fourth power, which means it does not have any real roots other than x=1x = 1, and it does not cross the x-axis.

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