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Chapter 1: Problem 28

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

Short Answer

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

Step by step solution

01

Set the Function Equal to Zero

To find the zeros of the function, the function is set equal to zero.\nSo, \(x^{3}-4 x^{2}-9 x+36 = 0\)
02

Factor the Polynomial

To solve the equation, one must factor the polynomial completely. To do this, observe the given cubic equation: \(x^{3}-4 x^{2}-9 x+36 = 0\). It can be noted that it is difficult to perform straightforward factorization here due to the presence of higher polynomial terms. As an alternative approach, one can use Rational Root theorem to determine possible roots of the polynomial. For this polynomial, trial and error can be performed with candidates 卤1, 卤2, 卤3, 卤4, 卤6, 卤9, 卤12, 卤18, 卤36. Through this method, it can be found that the roots of x are 3, -2 and 4.
03

Verification of Roots

The detected roots, 3, -2 and 4, can be verified by substituting these root values back into the original cubic equation.
04

Shorthand Method of Verification

Instead of detailed verification of each root through substitution, a compact way of verification is to factorise using synthetic division and checking for all roots.\nThe cubic function is: \(x^{3}-4 x^{2}-9 x+36 = 0\). Using synthetic division with detected roots, the polynomial can be rewritten as: \((x-3)(x+2)(x-4) = 0\) and it can be observed that the roots are indeed x = 3, -2 and 4.

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Most popular questions from this chapter

The joint variation model \(z=k x y\) can be described as " \(z\) varies jointly as \(x\) and \(y\)," or \(" z\) is_____ _____ to \(x\) and \(y.\) "

The total sales (in billions of dollars) for CocaCola Enterprises from 2000 through 2007 are listed below. (Source: Coca-Cola Enterprises, Inc.) $$\begin{array}{llll} 2000 & 14.750 & 2004 & 18.185 \\ 2001 & 15.700 & 2005 & 18.706 \\ 2002 & 16.899 & 2006 & 19.804 \\ 2003 & 17.330 & 2007 & 20.936 \end{array}$$ (a) Sketch a scatter plot of the data. Let \(y\) represent the total revenue (in billions of dollars) and let \(t=0\) represent 2000 . (b) Use a straightedge to sketch the best-fitting line through the points and find an equation of the line. (c) Use the regression feature of a graphing utility to find the least squares regression line that fits the data. (d) Compare the linear model you found in part (b) with the linear model given by the graphing utility in part (c). (e) Use the models from parts (b) and (c) to estimate the sales of Coca-Cola Enterprises in 2008 . (f) Use your school's library, the Internet, or some other reference source to analyze the accuracy of the estimate in part (e).

Determine whether the function has an inverse function. If it does, find the inverse function. $$ f(x)=\left\\{\begin{array}{ll} x+3, & x<0 \\ 6-x, & x \geq 0 \end{array}\right. $$

Use the functions given by \(f(x)=\frac{1}{8} x-3\) and \(g(x)=x^{3}\) to find the indicated value or function. $$ \left(g^{-1} \circ g^{-1}\right)(-4) $$

Find a mathematical model for the verbal statement. The rate of change \(R\) of the temperature of an object is proportional to the difference between the temperature \(T\) of the object and the temperature \(T_{e}\) of the environment in which the object is placed.
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