Problem 77 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the rational zeros of the polynomial function. $$P(x)=x^{4}-\frac{25}{4} x^{2}+9=\frac{1}{4}\left(4 x^{4}-25 x^{2}+36\right)$$

Step-by-Step Solution

Answer

The rational roots of the polynomial function are \{-\frac{3}{2}, \frac{3}{2}, -2, 2\}.

Step 1: Putting the Polynomial expression in quadratic form

By changing the variable, let's say \(y = x^2\), we can express the given polynomial in the quadratic form as follows: \(4y^2 - 25y + 36 = 0\).

Step 2: Factoring the quadratic form

Now the quadratic equation can be factored: \(4y^2 - 25y + 36 = 0\) which will give us \((4y - 9)(y - 4) = 0\)

Step 3: Finding roots for y

Without fail, y should equate to 0, this will yield two possible values for y: \(y1 = \frac{9}{4}\) and \(y2 = 4\). Since y = x^2, then x = ±√y.

Step 4: Finding values for x

For each value of y, we calculate the related values for x. For \(y1 = \frac{9}{4}\) we will have \(x1 = ±\frac{3}{2}\) and for \(y2 = 4\) we will have \(x2 = ±2\). Hence, the rational roots of the polynomial function \(P(x) = x^4 - \frac{25}{4}x^2 + 9\) are \(\{-\frac{3}{2}, \frac{3}{2}, -2, 2\}\)