If the function \(f(x) = 2x^3 - 9x^2 + 12x - 4\), find \(f(2)\).
["Title: How to Evaluate a Polynomial Function: Finding ( f(2) ) for ( f(x) = 2x^3 - 9x^2 + 12x - 4 )", "---", "Introduction", "Evaluating a polynomial function at a specific value is a fundamental skill in algebra and calculus. In this article, we’ll walk through a clear, step-by-step process to find ( f(2) ) for the function\n[ f(x) = 2x^3 - 9x^2 + 12x - 4. ]\nUnderstanding how to compute function values helps in analyzing graphs, solving real-world problems, and building more advanced mathematical concepts.", "---", "What Is the Function?", "We are given the cubic polynomial:\n[ f(x) = 2x^3 - 9x^2 + 12x - 4 ]\nThis function outputs a value for any real number input ( x ). Our goal is to determine what ( f ) returns when ( x = 2 ).", "---", "Step-by-Step Evaluation of ( f(2) )", "To compute ( f(2) ), substitute ( x = 2 ) into the function and simplify:", "[\n\begin{align}\nf(2) &= 2(2)^3 - 9(2)^2 + 12(2) - 4 \\n&= 2(8) - 9(4) + 12(2) - 4 \\n&= 16 - 36 + 24 - 4\n\end{align}\n]", "Now simplify step-by-step:", "[\n16 - 36 = -20\n]\n[\n-20 + 24 = 4\n]\n[\n4 - 4 = 0\n]", "Thus,\n[\nf(2) = 0\n]", "---", "Interpretation and Insight", "The result ( f(2) = 0 ) tells us that ( x = 2 ) is a root of the function — a value where the curve crosses or touches the x-axis. This is significant in algebra, especially when factoring polynomials or solving equations.", "Additionally, using the Remainder Theorem, since ( f(2) = 0 ), we know that ( (x - 2) ) is a factor of ( f(x) ), making polynomial division or factorization simpler.", "---", "Conclusion", "Evaluating functions like ( f(x) = 2x^3 - 9x^2 + 12x - 4 ) is straightforward once you substitute the value and simplify correctly. For ( x = 2 ), we found:\n[\nf(2) = 0\n]\nThis result confirms ( x = 2 ) is a root of the polynomial, offering valuable insight for solving equations and graphing.", "---", "Key Takeaway:\nWhen evaluating any polynomial at a specific ( x ), carefully substitute, compute powers and products accurately, and simplify step-by-step. Confirming roots via ( f(a) = 0 ) enhances your function analysis and problem-solving toolkit.", "---", "Keywords:\n( f(2) ), polynomial evaluation, evaluate polynomial, ( 2x^3 - 9x^2 + 12x - 4 ), function calculator, step-by-step math, algebra tips, root of a function, Remainder Theorem, polynomial roots", "Meta Description:\nLearn how to find ( f(2) ) for the cubic function ( f(x) = 2x^3 - 9x^2 + 12x - 4 ) step-by-step. Understand root finding, simplification, and interpretation — perfect for algebra students."]
