Answer:
[tex]p(x)=x^3-7x^2-5x+35[/tex]
Step-by-step explanation:
The required cubic equation has roots [tex]x=\sqrt{5} ,x=-\sqrt{5}[/tex] and [tex]x=7[/tex]
This implies that the cubic function has roots [tex](x-\sqrt{5} )[/tex], [tex](x+\sqrt{5} )[/tex] and [tex](x-7)[/tex].
Let p(x) be the cubic function, then we can write the factored form as:
[tex]p(x)=(x-\sqrt{5})(x+\sqrt{5})(x-7)[/tex]
We apply difference of two squares to the first two factors to get:
[tex]p(x)=(x^2-(\sqrt{5})^2)(x-7)[/tex]
[tex]p(x)=(x^2-5)(x-7)[/tex]
We expand using the distributive property to get:
[tex]p(x)=x^2(x-7)-5(x-7)[/tex]
[tex]p(x)=x^3-7x^2-5x+35[/tex]
