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This question I havent been able to understand. I am self studying and have no teacher to ask so I have been stuck for the past hour and don't get this is the full question the part I don't get is 14 c

enter image description here Any help is much appreciated. Thank you!

mary james
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  • Your screen shot is blurry and a bit dim. Please re-do with a clearly focused and well lit version. – coffeemath Feb 17 '21 at 16:07
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    Because a general formula to find the roots of a cubic exists. See https://en.wikipedia.org/wiki/Cubic_equation#General_cubic_formula – Joshua Wang Feb 17 '21 at 16:07
  • If you know one of the roots $r$, you can divide the polynomial by $x-r$ to get a quadratic with the other two roots, which you can then easily find. – Joshua Wang Feb 17 '21 at 16:07
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    @coffeemath i fixed it do you think you can look at it – mary james Feb 17 '21 at 16:12

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Hint: If a cubic polynomial $f(x)$ has exactly two roots, then one of them must be a double root, so that it is also a root of the derivative $f'(x)$, which is a quadratic polynomial.

Hans Lundmark
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  • I am confused so this is the answer. – mary james Feb 17 '21 at 16:28
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    @maryjames: Try it! Compute $f'(x)$ and solve the quadratic equation $f'(x)=0$. One of the two solutions you get should also be a (double) root to $f(x)=0$, and then you should be able to find the other root easily. – Hans Lundmark Feb 17 '21 at 17:05
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For part (a) of the problem: $p(x)=20x^3+8x^2-7x-3$ then $p(.5)=-2$ and $p(1)=+18.$ So since the sign is opposite at $.5$ and $1,$ there must be a root between them.

coffeemath
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