Question

Write a polynomial of degree 3 whose only roots are x=2 and x=2/5. Is there another polynomial of degree 3 that has the same roots?

Answer 1

One such polynomial might be

$(x-2)\left(x-\dfrac25\right)^2$

Another with the same roots would be

$(x-2)^2\left(x-\dfrac25\right)$

Answer 2

Answer:  The answer is $5x^3-22x^2+28x-8$ and $25x^3-70x^244x-8.$

Step-by-step explanation:  We are given to write a polynomial of degree 3 whose only roots are 2 and $\dfrac{2}{5}.$ Also, we need to find another polynomial of degree 3, if exists, that has the same roots.

The polynomial will be given by

$p(x)=(x-2)(x-2)(x-\dfrac{2}{5})\\\\\Rightarrow 5p(x)=(x^2-4x+4)(5x-2)\\\\\Rightarrow P(x)=5x^3-20x^2+20x-2x^2+8x-8\\\\\Rightarrow P(x)=5x^3-22x^2+28x-8.$

The other polynomial with same roots exist and is given by

$q(x)=(x-2)(x-\dfrac{2}{5})(x-\dfrac{2}{5})\\\\\Rightarrow 25q(x)=(x-2)(5x-2)^2\\\\\Rightarrow Q(x)=(x-2)(25x^2-20x+4)\\\\\Rightarrow Q(x)=25x^3-20x^2+4x-50x^2+40x-8\\\\\Rightarrow Q(x)=25x^3-70x^244x-8.$