Polynomial Functions

Question

Find a polynomial function that has the following zeros.

A. -3, 2, 1 B. -4, 0, 1, 2 C. + 1, + square root 2

D. 0, 2, 5 E. 2, 1+ square root 3 F. + 4, 0, + square root 2

G. -2, -1, 0, 1, 2 H. 1+ square root 2, + square root 3 I. 0, -3

Answer 1

For question B: Roots are -4, 0, 1, 2  

Write the roots as a factor:

Root -4 as a factor: (x-(-4)) = (x+4)

Root 0 as a factor: (x-0) = (x)

Root 1 as a factor: (x-1)

Root root 2 as factor: (x-2)

Multiply the factors together using foil: 

(x+4)(x)(x-1)(x-2) 

(x^2 + 4x)(x^2 - 3x + 2) 

 x^4 - 3x^3 + 2x^2 + 4x^3 -12x^2 +8x

Combine like terms 

x^4 + x^3 - 10x^2 + 8x

Answer: f(x) = x^4 + x^3 - 10x^2 + 8x 

Answer 2

These are all worked the same way. I will do problem H for you. You try the rest. Assuming the multiplicity is one for each zero,

f(x) = A [x - (1 + √2)] [x - (1 - √2)] (x - √3) (x + √3)

where A is real and A ≠ 0. You can expand to standard form if you wish.