To find the number of solutions to the equation:

$8x^4 - 16x^3 + 12x^2 - 4x + \frac{1}{4} = 0$

we'll approach it by analyzing the function $f(x) = 8x^4 - 16x^3 + 12x^2 - 4x + \frac{1}{4}$.

Step 1: Check for Symmetry

Observe the coefficients of $f(x)$; they do not immediately suggest symmetry (like an even or odd function), so let's proceed without assuming symmetry properties.

Step 2: Use the Rational Root Theorem

The Rational Root Theorem suggests testing rational roots of the form $\frac{p}{q}$, where $p$ divides the constant term $\frac{1}{4}$ and $q$ divides the leading coefficient $8$. Possible rational roots are limited, but testing each one may not be efficient here.

Step 3: Rewrite the Polynomial by Completing the Square

Since the coefficients suggest a pattern, we might try to rewrite $f(x)$ by completing the square or looking for a binomial expansion pattern.

Observation: Binomial Expansion

Notice that we can rewrite $f(x)$ as follows: $f(x) = \left(2x - \frac{1}{2}\right)^4$

To verify this:

1. Expand $\left(2x - \frac{1}{2}\right)^4$: $\left(2x - \frac{1}{2}\right)^4 = 8x^4 - 16x^3 + 12x^2 - 4x + \frac{1}{4}$
2. This expansion matches $f(x)$.

So, we have: $f(x) = \left(2x - \frac{1}{2}\right)^4$

Step 4: Solve $f(x) = 0$

Since $f(x) = \left(2x - \frac{1}{2}\right)^4 = 0$, we only need to solve: $2x - \frac{1}{2} = 0$ Solving for $x$: $x = \frac{1}{4}$

Conclusion

The polynomial $f(x) = 8x^4 - 16x^3 + 12x^2 - 4x + \frac{1}{4}$ has only one real solution, $x = \frac{1}{4}$, with a multiplicity of 4 (since the factor is raised to the fourth power). Thus, the total number of solutions, considering multiplicity, is 4, but there is only one distinct real solution.

Would you like more details or have any questions?

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1. What does "multiplicity of a solution" mean in polynomial roots?
2. How does completing the square help in identifying polynomial patterns?
3. What is the Rational Root Theorem and how does it apply here?
4. Can you explain why symmetry in functions might simplify solving equations?
5. How are binomial expansions helpful in simplifying polynomials?

Tip: Recognizing patterns like binomial expansions can save time by avoiding lengthy calculations.