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Question

# Determine if each of the following statements are true or false. Provide a brief explanation to justify each of your answers. a. The roots of a polynomial function are rational numbers. b. The process of finding zeros of a polynomial function is also referenced as finding the roots of the polynomial function. c. We can apply the zero-product property to determine the zeros of a polynomial function in factored form. d. The function used to model the box problem in Module 3 has two roots, at x = 0 and x = 4.25. e. The zeros of any function f represent the value(s) of x that when input into the function f return a value of 1 for the output variable f(x). f. The graph of a polynomial function is always a smooth curve, g. Polynomial functions are not always continuous.

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$\textbf{(a).}$ The roots of a polynomial functions are rational numbers.

This is a false statement because it is not necessary that the roots of a polynomial function are always rational numbers. Let us take a counterexample of polynomial function $f(x)=x^2+2x-1$ which has irrational roots.

To find the roots of polynomial function $f(x)=x^2+2x-1$, put $f(x)=0$ and solve for $x$ that means we have to solve the equation $x^2+2x-1=0$.

Let us solve this equation by using the quadratic formula.

According to the quadratic formula, the roots of equation $ax^2+bx+c=0$ are,

$x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}$

In the equation $x^2+2x-1=0$, we have $a=1$, $b=2$ and $c=-1$, therefore, the roots of the polynomial are,

\begin{align*} x&=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}\\ x&=\dfrac{-2\pm\sqrt{2^2-4\cdot1\cdot(-1)}}{2\cdot1}\\ x&=\dfrac{-2\pm\sqrt{4+4}}{2}\\ x&=\dfrac{-2\pm\sqrt{8}}{2}\\ x&=\dfrac{-2\pm\sqrt{2^2\cdot2}}{2}\\ x&=\dfrac{-2\pm2\sqrt{2}}{2}\\ x&=-1\pm\sqrt{2} \end{align*}

Thus, the roots of polynomial function $f(x)=x^2+2x-1$ are irrational that is $x=-1+\sqrt{2}$ and $x=-1-\sqrt{2}$.

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