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Chapter 2: Problem 10

Determine the number of zeros of the polynomial function. $$f(x)=x^{2}+5 x-6$$

Short Answer

Expert verified
The polynomial function \(f(x) = x^{2} +5x - 6\) has two zeros which are x = 1 and x = -6.

Step by step solution

01

Set the function equal to zero

Start by setting the polynomial function given, which is \(f(x)=x^{2}+5 x-6\), equal to zero. This translates to \(0 = x^{2} + 5x - 6\). This equation will assist in determining the roots of the equation that are essentially the zeros of the function.
02

Apply the Quadratic Formula

The Quadratic Formula is given by \(-b \pm \sqrt{b^2 - 4ac} \over 2a\), where a, b, and c are constants. For the equation \(0 = x^{2} + 5x - 6\), a = 1, b = 5, and c = -6. Substituting these values into the formula, we obtain \(x = \frac{-5 \pm \sqrt{5^2 - 4*1*(-6)}}{2*1} = \frac{-5 \pm \sqrt{5^2 - 4*1*(-6)}}{2}\). This simplifies to \(x = \frac{-5 \pm \sqrt{25 + 24}}{2} = \frac{-5 \pm \sqrt{49}}{2}\), which further simplifies to \(x = \frac{-5 \pm 7}{2}\). Thus, two possible solutions are \(x = \frac{-5 + 7} {2} = 1\) and \(x = \frac{-5 - 7} {2} = -6\)
03

Concluding the number of zeros

Since there are two solutions to the equation, it follows that the polynomial function \(f(x) = x^{2} +5x - 6\) has two zeros. They are x = 1 and x = -6.

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Most popular questions from this chapter

Complete the following. \begin{aligned} &i^{1}=i \quad i^{2}=-1 \quad i^{3}=-i \quad i^{4}=1\\\ &i^{5}=\quad i^{6}=\quad i^{7}=\quad i^{8}=\\\ &i^{9}=\\\ &i^{10}=\quad i^{11}=\quad i^{12}= \end{aligned} What pattern do you see? Write a brief description of how you would find \(i\) raised to any positive integer power.

Think About It Let \(y=f(x)\) be a cubic polynomial with leading coefficient \(a=-1\) and \(f(2)=f(i)=0\) Write an equation for \(f\)

Find all the zeros of the function. When there is an extended list of possible rational zeros, use a graphing utility to graph the function in order to disregard any of the possible rational zeros that are obviously not zeros of the function. $$f(x)=16 x^{3}-20 x^{2}-4 x+15$$

For each function, identify the degree of the function and whether the degree of the function is even or odd. Identify the leading coefficient and whether the leading coefficient is positive or negative. Use a graphing utility to graph each function. Describe the relationship between the degree of the function and the sign of the leading coefficient of the function and the right-hand and left-hand behavior of the graph of the function. (a) \(f(x)=x^{3}-2 x^{2}-x+1\) (b) \(f(x)=2 x^{5}+2 x^{2}-5 x+1\) (c) \(f(x)=-2 x^{5}-x^{2}+5 x+3\) (d) \(f(x)=-x^{3}+5 x-2\) (e) \(f(x)=2 x^{2}+3 x-4\) (f) \(f(x)=x^{4}-3 x^{2}+2 x-1\) (g) \(f(x)=x^{2}+3 x+2\)

Write the polynomial (a) as the product of factors that are irreducible over the rationals, (b) as the product of linear and quadratic factors that are irreducible over the reals, and (c) in completely factored form. \(f(x)=x^{4}-3 x^{3}-x^{2}-12 x-20\) (Hint: One factor is \(\left.x^{2}+4 .\right)\)
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