Find the roots of the polynomial function \(f(x) = x^3 - 6x^2 + 9x - 4\) using the Factor Theorem.
Answer
Step 3: Now, we set \(x^2 - 5x + 4 = 0\) and solve for \(x\). This can be factored as \((x - 1)(x - 4) = 0\). Setting each factor equal to zero gives the roots \(x = 1\) and \(x = 4\).
Step 1 ：Step 1: According to the Factor Theorem, if \(f(a) = 0\), then \(x - a\) is a factor of the polynomial. We start by testing values to find a root of the polynomial. Plugging in \(x = 1\), we get \(f(1) = 1 - 6 + 9 - 4 = 0\). So, \(x = 1\) is a root of the polynomial and \(x - 1\) is a factor of the polynomial.
Step 2 ：Step 2: Now we divide the polynomial \(f(x)\) by the factor \(x - 1\) to find the other factors. Performing the synthetic division, we get the quotient \(x^2 - 5x + 4\).
Step 3 ：Step 3: Now, we set \(x^2 - 5x + 4 = 0\) and solve for \(x\). This can be factored as \((x - 1)(x - 4) = 0\). Setting each factor equal to zero gives the roots \(x = 1\) and \(x = 4\).
