Show instructions Question 1 (1 point) Using technology, find the roots of the polynomial equation: \[ f(x)=2 x^{2}+8.5 x+6.5 \] a \( \quad-\frac{2}{9},-1 \) b \( -\frac{13}{4},-1 \) c \( \frac{9}{8}, 17 \) d \( -\frac{2}{9}, 8 \) Question 2 (1 point) Using the rational roots theorem, list all the possible rational roots of \( f(x)=x^{3}-4 x^{2}+8 x-5 \) a \( \quad-5,-1,1,5 \) b \( \quad-5,-3-1,1,3,5 \) c \( -8,-5,-1,1,5,8 \) d none of the above Question 3 (1 point)

**Question 1** We are given the equation \[ f(x)=2x^2+8.5x+6.5=0. \] We use the quadratic formula \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}, \] with \( a=2 \), \( b=8.5 \), and \( c=6.5 \). 1. Compute the discriminant: \[ b^2 - 4ac = (8.5)^2 - 4(2)(6.5) = 72.25 - 52 = 20.25. \] 2. Take the square root: \[ \sqrt{20.25} = 4.5. \] 3. Solve for the roots: \[ x = \frac{-8.5 \pm 4.5}{4}. \] - For the \(+\) sign: \[ x = \frac{-8.5+4.5}{4} = \frac{-4}{4} = -1. \] - For the \(-\) sign: \[ x = \frac{-8.5-4.5}{4} = \frac{-13}{4} = -\frac{13}{4}. \] Thus, the roots are \( -\frac{13}{4} \) and \( -1 \), which corresponds to option **b**. --- **Question 2** The polynomial given is \[ f(x)=x^3-4x^2+8x-5. \] The Rational Root Theorem tells us that any rational root, in reduced form, will be of the form \[ \frac{p}{q}, \] where \( p \) is a factor of the constant term \(-5\) and \( q \) is a factor of the leading coefficient \(1\). - Factors of \(-5\) are \( \pm1, \pm5 \). - Factors of \(1\) are \( \pm1 \). Thus, the possible rational roots are \[ \pm1, \pm5. \] This corresponds to option **a**. --- **Question 3** No content or details were provided for Question 3, so it is not possible to solve or list an answer for this question.