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)

To find the roots of the polynomial equation \( f(x)=2x^2+8.5x+6.5 \), you can use technology like a graphing calculator or online software such as Desmos or Wolfram Alpha. By inputting the polynomial, you’ll quickly find the values of \( x \) where \( f(x) = 0 \). For this particular quadratic, the roots simplify down to approximately \( -1.45 \) and \( -2.25 \), which don't exactly match the provided options. It's like trying to win a game by landing on a square just a bit outside of your reach! As for the rational roots theorem, it states that any rational solution of the polynomial equation \( f(x) \) must be a factor of the constant term divided by a factor of the leading coefficient. For \( f(x)=x^3 - 4x^2 + 8x - 5 \), the possible rational roots would include factors of \(-5\) (the constant term) over factors of \(1\) (the leading coeff.). So, the possible rational roots to consider are \( \pm 1, \pm 5 \). Careful consideration here might lead you to option \( a \), which lists these potential roots, giving you a good chance at making an informed guess!