\(\displaystyle \begin{align*}&\text{The function }f(x)=286x^{2} - 43x - 204\text{ crosses the x-axis}\\&\text{when it has a value of zero. To find the x values that}\\&\text{satisfy this, we can either factor the equation:}\\&(22x + 17)\cdot (13x - 12)\\&\text{or, if this is not intuitive, use the quadratic formula:}\\&\frac{-b\pm\sqrt{b^2-4ac}}{2a}(\text{for equations of the form: }a^2x+bx+c)\\&\frac{-(-43)\pm\sqrt{(-43)^2-4(286)(-204)}}{2(286)}=\frac{43\pm485}{572}\\&\text{Either way, we find thatthe function crosses the x-axis at}\ x=\frac{12}{13}\text{ and }-\frac{17}{22}.\end{align*}\)

\(\displaystyle \begin{align*}&\text{The function }f(x)=288x - 175x^{2} + 256\text{ crosses the x-axis}\\&\text{when it has a value of zero. To find the x values that}\\&\text{satisfy this, we can either factor the equation:}\\&-(25x + 16)\cdot (7x - 16)\\&\text{or, if this is not intuitive, use the quadratic formula:}\\&\frac{-b\pm\sqrt{b^2-4ac}}{2a}(\text{for equations of the form: }a^2x+bx+c)\\&\frac{-(288)\pm\sqrt{(288)^2-4(-175)(256)}}{2(-175)}=\frac{-288\pm512}{-350}\\&\text{Either way, we find thatthe function crosses the x-axis at} \ x=\frac{16}{7}\text{ and }-\frac{16}{25}.\end{align*}\)

\(\displaystyle \begin{align*}&\text{The function }f(x)=- 154x - 35x^{2} - 147\text{ crosses the x-axis}\\&\text{when it has a value of zero. To find the x values that}\\&\text{satisfy this, we can either factor the equation:}\\&-7\cdot (x + 3)\cdot (5x + 7)\\&\text{or, if this is not intuitive, use the quadratic formula:}\\&\frac{-b\pm\sqrt{b^2-4ac}}{2a}(\text{for equations of the form: }a^2x+bx+c)\\&\frac{-(-154)\pm\sqrt{(-154)^2-4(-35)(-147)}}{2(-35)}=\frac{154\pm56}{-70}\\&\text{Either way, we find thatthe function crosses the x-axis at}\ x=-3\text{ and }-\frac{7}{5}.\end{align*}\)

\(\displaystyle \begin{align*}&\text{The function }f(x)=15 - 76x^{2} - 281x\text{ crosses the x-axis}\\&\text{when it has a value of zero. To find the x values that}\\&\text{satisfy this, we can either factor the equation:}\\&-(4x + 15)\cdot (19x - 1)\\&\text{or, if this is not intuitive, use the quadratic formula:}\\&\frac{-b\pm\sqrt{b^2-4ac}}{2a}(\text{for equations of the form: }a^2x+bx+c)\\&\frac{-(-281)\pm\sqrt{(-281)^2-4(-76)(15)}}{2(-76)}=\frac{281\pm289}{-152}\\&\text{Either way, we find thatthe function crosses the x-axis at} \ x=\frac{1}{19}\text{ and }-\frac{15}{4}.\end{align*}\)

\(\displaystyle \begin{align*}&\text{The function }f(x)=- 435x - 84x^{2} - 450\text{ crosses the x-axis}\\&\text{when it has a value of zero. To find the x values that}\\&\text{satisfy this, we can either factor the equation:}\\&-3\cdot (4x + 15)\cdot (7x + 10)\\&\text{or, if this is not intuitive, use the quadratic formula:}\\&\frac{-b\pm\sqrt{b^2-4ac}}{2a}(\text{for equations of the form: }a^2x+bx+c)\\&\frac{-(-435)\pm\sqrt{(-435)^2-4(-84)(-450)}}{2(-84)}=\frac{435\pm195}{-168}\\&\text{Either way, we find thatthe function crosses the x-axis at}\ x=-\frac{15}{4}\text{ and }-\frac{10}{7}.\end{align*}\)

\(\displaystyle \begin{align*}&\text{The function }f(x)=199x + 200x^{2} - 28\text{ crosses the x-axis}\\&\text{when it has a value of zero. To find the x values that}\\&\text{satisfy this, we can either factor the equation:}\\&(25x + 28)\cdot (8x - 1)\\&\text{or, if this is not intuitive, use the quadratic formula:}\\&\frac{-b\pm\sqrt{b^2-4ac}}{2a}(\text{for equations of the form: }a^2x+bx+c)\\&\frac{-(199)\pm\sqrt{(199)^2-4(200)(-28)}}{2(200)}=\frac{-199\pm249}{400}\\&\text{Either way, we find thatthe function crosses the x-axis at}\ x=\frac{1}{8}\text{ and }-\frac{28}{25}.\end{align*}\)

\(\displaystyle \begin{align*}&\text{The function }f(x)=348x + 96x^{2} + 270\text{ crosses the x-axis}\\&\text{when it has a value of zero. To find the x values that}\\&\text{satisfy this, we can either factor the equation:}\\&6\cdot (2x + 5)\cdot (8x + 9)\\&\text{or, if this is not intuitive, use the quadratic formula:}\\&\frac{-b\pm\sqrt{b^2-4ac}}{2a}(\text{for equations of the form: }a^2x+bx+c)\\&\frac{-(348)\pm\sqrt{(348)^2-4(96)(270)}}{2(96)}=\frac{-348\pm132}{192}\\&\text{Either way, we find thatthe function crosses the x-axis at} \ x=-\frac{5}{2}\text{ and }-\frac{9}{8}.\end{align*}\)

\(\displaystyle \begin{align*}&\text{The function }f(x)=251x - 33x^{2} - 400\text{ crosses the x-axis}\\&\text{when it has a value of zero. To find the x values that}\\&\text{satisfy this, we can either factor the equation:}\\&-(11x - 25)\cdot (3x - 16)\\&\text{or, if this is not intuitive, use the quadratic formula:}\\&\frac{-b\pm\sqrt{b^2-4ac}}{2a}(\text{for equations of the form: }a^2x+bx+c)\\&\frac{-(251)\pm\sqrt{(251)^2-4(-33)(-400)}}{2(-33)}=\frac{-251\pm101}{-66}\\&\text{Either way, we find thatthe function crosses the x-axis at}\ x=\frac{16}{3}\text{ and }\frac{25}{11}.\end{align*}\)

To find the values for \(\displaystyle x\) in which the polynomial equals \(\displaystyle 0\), we first want to factor the equation:

\(\displaystyle 2x^2-2x-15=0\)

\(\displaystyle x=\frac{-(-2)\pm \sqrt{(-2)^2-4(2)(-15)}}{2(2)}\)

\(\displaystyle \\x=\frac{2\pm \sqrt{4+120}}{4}\)

\(\displaystyle x=\frac{2\pm \sqrt{124}}{4}\)

\(\displaystyle x=\frac{2\pm 2\sqrt{31}}{4}\)

\(\displaystyle x=\frac{1\pm \sqrt{31}}{2}\)

By Descartes' Rule of Signs, the number of sign changes - changes from positive to negative coefficient signs - in \(\displaystyle P(x)\) gives the maximum number of positive real zeroes; the actual number of positive real zeroes must be that many or differ by an even number.

If the polynomial 

\(\displaystyle P(x) = 3x^{5}+ 6x^{3} + 17x^{2} + 7x + 10\) 

is examined, it can be seen that there are no changes in sign from term to term; all coefficients are positive. Therefore, there can be no positive real zeroes.