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?

One such polynomial might be

[tex](x-2)\left(x-\dfrac25\right)^2[/tex]

Another with the same roots would be

[tex](x-2)^2\left(x-\dfrac25\right)[/tex]

Answer Link

ColinJacobus ColinJacobus · Answer 2 · 2019-02-28T10:31:07+01:00

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

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

The polynomial will be given by

[tex]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.[/tex]

The other polynomial with same roots exist and is given by

[tex]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.[/tex]