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College Algebra : Zeros/Roots of a Polynomial

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Example Questions

Example Question #1 : Zeros/Roots Of A Polynomial

$\begin{align*}&\text{Find the value(s) of x where the function: }f(x)=- 297x - 270x^{2} - 27\\&\text{crosses the x-axis.}\end{align*}$

Possible Answers:

$x=1,\frac{1}{10}$

x=540,54

$x=-1,-\frac{1}{10}$

x=-540,-54

Correct answer:

$x=-1,-\frac{1}{10}$

Explanation:

$\begin{align*}&\text{The function }f(x)=- 297x - 270x^{2} - 27\text{ crosses the x-axis}\\&\text{when it has a value of zero. To find the x values that}\\&\text{satisfy this, we can either factor the equation:}\\&-27\cdot (x + 1)\cdot (10x + 1)\\&\text{or, if this is not intuitive, use the quadratic formula:}\\&\frac{-b\pm\sqrt{b^2-4ac}}{2a}(\text{for equations of the form: }a^2x+bx+c)\\&\frac{-(-297)\pm\sqrt{(-297)^2-4(-270)(-27)}}{2(-270)}=\frac{297\pm243}{-540}\\&\text{Either way, we find that the function crosses the x-axis at}\ x=-1\text{ and }-\frac{1}{10}.\end{align*}$

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Example Question #2 : Zeros/Roots Of A Polynomial

$\begin{align*}&\text{Find the value(s) of x where the function: }f(x)=- 822x - 288x^{2} - 567\\&\text{has a zero value.}\end{align*}$

Possible Answers:

$x=-\frac{7}{6},-\frac{27}{16}$

x=972,672

$x=\frac{7}{6},\frac{27}{16}$

x=-972,-672

Correct answer:

$x=-\frac{7}{6},-\frac{27}{16}$

Explanation:

$\begin{align*}&\text{Given }f(x)=- 822x - 288x^{2} - 567\text{, to find when it has a value of}\\&\text{zero, we need a method to determine the x values that}\\&\text{make this happen. We can either factor the equation:}\\&-3\cdot (16x + 27)\cdot (6x + 7)\\&\text{or, if this is not intuitive, use the quadratic formula:}\\&\frac{-b\pm\sqrt{b^2-4ac}}{2a}(\text{for equations of the form: }a^2x+bx+c)\\&\frac{-(-822)\pm\sqrt{(-822)^2-4(-288)(-567)}}{2(-288)}=\frac{822\pm150}{-576}\\&\text{Either way, we find thatthe function crosses the x-axis at}\ x=-\frac{7}{6}\text{ and }-\frac{27}{16}.\end{align*}$

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Example Question #3 : Zeros/Roots Of A Polynomial

$\begin{align*}&\text{Find the value(s) of x where the function: }f(x)=- 648x - 152x^{2} - 576\\&\text{crosses the x-axis.}\end{align*}$

Possible Answers:

x=-912,-384

$x=3,\frac{24}{19}$

$x=-3,-\frac{24}{19}$

x=912,384

Correct answer:

$x=-3,-\frac{24}{19}$

Explanation:

$\begin{align*}&\text{The function }f(x)=- 648x - 152x^{2} - 576\text{ crosses the x-axis}\\&\text{when it has a value of zero. To find the x values that}\\&\text{satisfy this, we can either factor the equation:}\\&-8\cdot (x + 3)\cdot (19x + 24)\\&\text{or, if this is not intuitive, use the quadratic formula:}\\&\frac{-b\pm\sqrt{b^2-4ac}}{2a}(\text{for equations of the form: }a^2x+bx+c)\\&\frac{-(-648)\pm\sqrt{(-648)^2-4(-152)(-576)}}{2(-152)}=\frac{648\pm264}{-304}\\&\text{Either way, we find thatthe function crosses the x-axis at} \ x=-3\text{ and }-\frac{24}{19}.\end{align*}$

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Example Question #4 : Zeros/Roots Of A Polynomial

$\begin{align*}&\text{Find the value(s) of x where the function: }f(x)=712x - 110x^{2} - 728\\&\text{crosses the x-axis.}\end{align*}$

Possible Answers:

$x=-\frac{26}{5},-\frac{14}{11}$

x=-280,-1144

x=280,1144

$x=\frac{26}{5},\frac{14}{11}$

Correct answer:

$x=\frac{26}{5},\frac{14}{11}$

Explanation:

$\begin{align*}&\text{The function }f(x)=712x - 110x^{2} - 728\text{ crosses the x-axis}\\&\text{when it has a value of zero. To find the x values that}\\&\text{satisfy this, we can either factor the equation:}\\&-2\cdot (11x - 14)\cdot (5x - 26)\\&\text{or, if this is not intuitive, use the quadratic formula:}\\&\frac{-b\pm\sqrt{b^2-4ac}}{2a}(\text{for equations of the form: }a^2x+bx+c)\\&\frac{-(712)\pm\sqrt{(712)^2-4(-110)(-728)}}{2(-110)}=\frac{-712\pm432}{-220}\\&\text{Either way, we find thatthe function crosses the x-axis at}\ x=\frac{26}{5}\text{ and }\frac{14}{11}.\end{align*}$

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Example Question #1 : Zeros/Roots Of A Polynomial

$\begin{align*}&\text{Find the value(s) of x where the function: }f(x)=- 544x - 152x^{2} - 480\\&\text{crosses the x-axis.}\end{align*}$

Possible Answers:

$x=2,\frac{30}{19}$

x=608,480

$x=-2,-\frac{30}{19}$

x=-608,-480

Correct answer:

$x=-2,-\frac{30}{19}$

Explanation:

$\begin{align*}&\text{The function }f(x)=- 544x - 152x^{2} - 480\text{ crosses the x-axis}\\&\text{when it has a value of zero. To find the x values that}\\&\text{satisfy this, we can either factor the equation:}\\&-8\cdot (x + 2)\cdot (19x + 30)\\&\text{or, if this is not intuitive, use the quadratic formula:}\\&\frac{-b\pm\sqrt{b^2-4ac}}{2a}(\text{for equations of the form: }a^2x+bx+c)\\&\frac{-(-544)\pm\sqrt{(-544)^2-4(-152)(-480)}}{2(-152)}=\frac{544\pm64}{-304}\\&\text{Either way, we find thatthe function crosses the x-axis at}\ x=-2\text{ and }-\frac{30}{19}.\end{align*}$

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Example Question #6 : Zeros/Roots Of A Polynomial

$\begin{align*}&\text{Find the value(s) of x where the function: }f(x)=225x^{2} - 623x + 374\\&\text{crosses the x-axis.}\end{align*}$

Possible Answers:

x=850,396

$x=\frac{22}{25},\frac{17}{9}$

$x=-\frac{22}{25},-\frac{17}{9}$

x=-850,-396

Correct answer:

$x=\frac{22}{25},\frac{17}{9}$

Explanation:

$\begin{align*}&\text{The function }f(x)=225x^{2} - 623x + 374\text{ crosses the x-axis}\\&\text{when it has a value of zero. To find the x values that}\\&\text{satisfy this, we can either factor the equation:}\\&(25x - 22)\cdot (9x - 17)\\&\text{or, if this is not intuitive, use the quadratic formula:}\\&\frac{-b\pm\sqrt{b^2-4ac}}{2a}(\text{for equations of the form: }a^2x+bx+c)\\&\frac{-(-623)\pm\sqrt{(-623)^2-4(225)(374)}}{2(225)}=\frac{623\pm227}{450}\\&\text{Either way, we find thatthe function crosses the x-axis at}\ x=\frac{22}{25}\text{ and }\frac{17}{9}.\end{align*}$

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Example Question #1 : Zeros/Roots Of A Polynomial

$\begin{align*}&\text{Find the value(s) of x where the function: }f(x)=108x^{2} - 60x - 25\\&\text{has a zero value.}\end{align*}$

Possible Answers:

x=180,-60

$x=-\frac{5}{6},-\frac{5}{18}$

$x=\frac{5}{6},-\frac{5}{18}$

$x=-\frac{5}{6},\frac{5}{18}$

Correct answer:

$x=\frac{5}{6},-\frac{5}{18}$

Explanation:

$\begin{align*}&\text{Given }f(x)=108x^{2} - 60x - 25\text{, to find when it has a value of}\\&\text{zero, we need a method to determine the x values that}\\&\text{make this happen. We can either factor the equation:}\\&(18x + 5)\cdot (6x - 5)\\&\text{or, if this is not intuitive, use the quadratic formula:}\\&\frac{-b\pm\sqrt{b^2-4ac}}{2a}(\text{for equations of the form: }a^2x+bx+c)\\&\frac{-(-60)\pm\sqrt{(-60)^2-4(108)(-25)}}{2(108)}=\frac{60\pm120}{216}\\&\text{Either way, we find thatthe function crosses the x-axis at}\ x=\frac{5}{6}\text{ and }-\frac{5}{18}.\end{align*}$

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Example Question #8 : Zeros/Roots Of A Polynomial

$\begin{align*}&\text{Find the value(s) of x where the function: }f(x)=421x + 306x^{2} - 210\\&\text{has a zero value.}\end{align*}$

Possible Answers:

$x=\frac{7}{18},-\frac{30}{17}$

$x=-\frac{7}{18},\frac{30}{17}$

x=238,-1080

x=-238,1080

Correct answer:

$x=\frac{7}{18},-\frac{30}{17}$

Explanation:

$\begin{align*}&\text{Given }f(x)=421x + 306x^{2} - 210\text{, to find when it has a value of}\\&\text{zero, we need a method to determine the x values that}\\&\text{make this happen. We can either factor the equation:}\\&(17x + 30)\cdot (18x - 7)\\&\text{or, if this is not intuitive, use the quadratic formula:}\\&\frac{-b\pm\sqrt{b^2-4ac}}{2a}(\text{for equations of the form: }a^2x+bx+c)\\&\frac{-(421)\pm\sqrt{(421)^2-4(306)(-210)}}{2(306)}=\frac{-421\pm659}{612}\\&\text{Either way, we find thatthe function crosses the x-axis at} \ x=\frac{7}{18}\text{ and }-\frac{30}{17}.\end{align*}$

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Example Question #9 : Zeros/Roots Of A Polynomial

$\begin{align*}&\text{Find the value(s) of x where the function: }f(x)=323x + 29x^{2} - 300\\&\text{has a zero value.}\end{align*}$

Possible Answers:

x=50,-696

x=-50,696

$x=-12,\frac{25}{29}$

$x=12,-\frac{25}{29}$

Correct answer:

$x=-12,\frac{25}{29}$

Explanation:

$\begin{align*}&\text{Given }f(x)=323x + 29x^{2} - 300\text{, to find when it has a value of}\\&\text{zero, we need a method to determine the x values that}\\&\text{make this happen. We can either factor the equation:}\\&(x + 12)\cdot (29x - 25)\\&\text{or, if this is not intuitive, use the quadratic formula:}\\&\frac{-b\pm\sqrt{b^2-4ac}}{2a}(\text{for equations of the form: }a^2x+bx+c)\\&\frac{-(323)\pm\sqrt{(323)^2-4(29)(-300)}}{2(29)}=\frac{-323\pm373}{58}\\&\text{Either way, we find thatthe function crosses the x-axis at}\ x=-12\text{ and }\frac{25}{29}.\end{align*}$

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Example Question #10 : Zeros/Roots Of A Polynomial

$\begin{align*}&\text{Find the root(s) the function: }f(x)=379x + 276x^{2} - 696\end{align*}$

Possible Answers:

x=576,-1334

$x=-\frac{24}{23},\frac{29}{12}$

x=-576,1334

$x=\frac{24}{23},-\frac{29}{12}$

Correct answer:

$x=\frac{24}{23},-\frac{29}{12}$

Explanation:

$\begin{align*}&\text{The roots of the function }f(x)=379x + 276x^{2} - 696\text{ are the x values}\\&\text{that give it a value of zero. To find these x values}\\&\text{we can either factor the equation:}\\&(12x + 29)\cdot (23x - 24)\\&\text{or, if this is not intuitive, use the quadratic formula:}\\&\frac{-b\pm\sqrt{b^2-4ac}}{2a}(\text{for equations of the form: }a^2x+bx+c)\\&\frac{-(379)\pm\sqrt{(379)^2-4(276)(-696)}}{2(276)}=\frac{-379\pm955}{552}\\&\text{Either way, we find thatthe function crosses the x-axis at}\ x=\frac{24}{23}\text{ and }-\frac{29}{12}.\end{align*}$

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College Algebra : Zeros/Roots of a Polynomial

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