Problem 79 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the rational zeros of the polynomial function. $$f(x)=x^{3}-\frac{1}{4} x^{2}-x+\frac{1}{4}=\frac{1}{4}\left(4 x^{3}-x^{2}-4 x+1\right)$$

Step-by-Step Solution

Answer

The rational zeros of the given polynomial function \(f(x)=\frac{1}{4}\left(4x^{3}-x^{2}-4x+1\right)\) are -1, 1 and \(\frac{1}{4}\).

Step 1: Identify coefficients and constant

Begin by identifying the leading coefficient and the constant term in the given polynomial. The leading coefficient of the polynomial is 4 (from the term \(4x^3\)) and the constant term is 1.

Step 2: Find factors

Next, find all the factors of the leading coefficient and the constant term. The factors of 4 are -4, -2, -1, 1, 2, 4 and the factors of 1 are -1 and 1.

Step 3: Form possible rational zeros

Now form possible rational zeros by considering all combinations of factors of the constant term divided by factors of the leading coefficient. This gives a list of possible rational zeros: \(-4,-2,-1,-\frac{1}{2},-\frac{1}{4},1,\frac{1}{2},\frac{1}{4},2,4\)

Step 4: Check for rational zeros

Each of these possibilities can be tested in the original polynomial to find the rational zeros. The rational zeros that would satisfy the polynomial are -1, 1 and \frac{1}{4}.