Question

Which of these polynomial equations is of least degree and has -1, 2, and 4 as three of its roots?

5x2 + 2x + 8 = 0
(x + 2)(x - 1)(x + 4) = 0
x3 - 5x2 + 2x + 8 = 0
x4 - x3 - 5x2 + 2x + 8 = 0

Answer 1

x^3 - 5 x^2 + 2 x + 8 = 0

Explanation:

Solve for x over the real numbers:

x^3 - 5 x^2 + 2 x + 8 = 0

The left hand side factors into a product with three terms:

(x - 4) (x - 2) (x + 1) = 0

Split into three equations:

x - 4 = 0 or x - 2 = 0 or x + 1 = 0

Add 4 to both sides:

x = 4 or x - 2 = 0 or x + 1 = 0

Add 2 to both sides:

x = 4 or x = 2 or x + 1 = 0

Subtract 1 from both sides:

Answer: x = 4 or x = 2 or x = -1

Answer 2

The polynomial equations which is of least degree and has -1, 2, and 4 as three of its roots is:

`x^3-5x^2+2x+8=0`

Explanation:

The three roots of a polynomial are given as:

-1, 2 and 4

We know that the least degree polynomial whose roots are a,b and c is given by the equation:

`(x-a)(x-b)(x-c)=0`

Here we have:

a= -1 , b=2 and c=4

Hence, the polynomial equation is given by:

`(x-(-1))(x-2)(x-4)=0\\\\\\i.e.\\\\\\(x+1)(x-2)(x-4)=0\\\\i.e.\\\\\\(x^2-x-2)(x-4)=0\\\\\\i.e.\\\\\\x^2(x-4)-x(x-4)-2(x-4)=0\\\\\\x^3-4x^2-x^2+4x-2x+8=0\\\\\\x^3-5x^2+2x+8=0`

Hence, the answer is:

`x^3-5x^2+2x+8=0`