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Chapter 1: Problem 21

Find the zeros of the function algebraically. $$f(x)=x^{3}-4 x^{2}-9 x+36$$

Short Answer

Expert verified
The roots of the function \(f(x)=x^{3}-4 x^{2}-9 x+36\) are x = -1, x = 3, x = 4.

Step by step solution

01

Simplify the equation

First, set the function equal to zero and simplify where possible. As such, the equation emerges as \(x^{3}-4 x^{2}-9 x+36 = 0\).
02

Check for factorization

Now let's check for possible factorization. None are immediately apparent, so we need to apply the Rational Root Theorem.
03

Apply the Rational Root Theorem

The Rational Root Theorem provides potential rational roots of the function, which are all possible ways to combine factors of the leading coefficient (here 1) and the constant term (here -36). The possibilities to check are \(卤1, 卤2, 卤3, 卤4, 卤6, 卤9, 卤12, 卤18, 卤36\). Values of x that satisfy the equation (i.e. make f(x) = 0) are roots of the equation. If some of these turn out to be roots, we can use them to factor the original polynomial.
04

Find the roots

By plugging in the possible values of x from Step 3 into the equation, we get the following three roots: \(x = -1, x = 3, x = 4\). Therefore, the polynomial factors as \( (x+1)(x-3)(x-4) = 0\).
05

Confirm the roots

Replace x with each of the roots in the original function to confirm that they return a value of f(x) = 0. Confirmation of these roots would validate them as the zeros of the function.

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Most popular questions from this chapter

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