Polynomial conditions

Question

Write a polynomial f(x) that meets the given conditions.

 

Degree 3 polynomial with leading coefficient of 1 and zeros of 1 , 3 and -5.

 

f(x)= 

Answer 1

If the zeros are 1, 3 and -5, then we know the factors of the this polynomial. 

 

We can set this up as 

(x-1) (x-3) (x+5)

 

Expanding this

(x² - 4x + 3)(x+5)

 

and

x³ - 4x² + 3x + 5x² - 20x + 15

 

Combining terms

x³ + x² - 17x + 15

 

This is a 3 degree polynomial (begins with x³) with a leading coefficient of 1 (The coefficient of the x³ term is 1).

 

Hope this helps.

Answer 2

Since the polynomial has degree 3, it can't have more than 3 distinct roots.  Also, if c is a root, then x-c is a factor.  

 

f(x) = (x-1)(x-3)(x-(-5))

 

      = (x-1)(x-3)(x+5) = (x-1)(x²+2x-15)

 

                               = x³+2x²-15x-x²-2x+15 = x³+x²-17x+15