I need to write a polynomial of the smallest degree with roots 5 and 6 and the answer can't have parentheses. Can someone point me in the right direction. I have several of these to do.
(x-5)(x-6)=f(x)
x^2-11x+30=f(x)
I came up with (x+5)(x+6) = 0
x^2 - 11x + 30 = 0
Would that also be correct?
Thank you for helping
I can answer my own question-NO yours makes more sense. Its the first day of these so I'm still alittle confused. Thank you for helping
To write a polynomial with roots 5 and 6 (let's call it "x"), you need to use the fact that if a number is a root of a polynomial, then it is a solution to the equation formed by setting the polynomial equal to zero.
Let's start by writing the equations:
x - 5 = 0 (since 5 is a root)
x - 6 = 0 (since 6 is a root)
To remove the parentheses, you can rewrite these equations without them:
x - 5 = 0 can be written as x + (-5) = 0
x - 6 = 0 can be written as x + (-6) = 0
Now, to find the polynomial with these roots, you multiply the factors together:
(x + (-5))(x + (-6))
To simplify, you can distribute:
(x - 5)(x - 6)
Now you have the polynomial with the smallest degree that has 5 and 6 as roots, without any parentheses: x² - 11x + 30
So the answer is x² - 11x + 30.